Facultad de Ciencias

Departamento de Matemáticas

A study on the singularities of the exponential maps of Riemann and Finsler
manifolds, with applications to Hamilton-Jacobi equations, and the Ambrose
conjecture

dedicatory
Tesis doctoral dirigida por
Luis Guijarro Santamaría __

###### Contents

  1. I Relation between Hamilton-Jacobi equations and Finsler Geometry
  2. II Singularities of the exponential map
  3. III Structure of the cut locus
  4. IV Characterization of the cut locus and the balanced split loci
  5. V The Ambrose conjecture
  6. VI Summary of results
  7. 1 Preliminaries
    1. 1.1 A little background
      1. 1.1.1 Approximate Tangent Cone
      2. 1.1.2 Subdifferentials of semiconcave functions
      3. 1.1.3 Duality in Finsler Geometry
    2. 1.2 Exponential maps of Finsler Manifolds
      1. 1.2.1 Regular exponential map
      2. 1.2.2 Special coordinates
    3. 1.3 The Cut Locus
    4. 1.4 Hamilton-Jacobi equations and Finsler geometry
      1. 1.4.1 Characteristics of the HJBVP
      2. 1.4.2 Viscosity solutions of Hamilton-Jacobi equations
  8. 2 A new way to look at Cut and Singular Loci
    1. 2.1 The relation between Finsler geometry and Hamilton-Jacobi BVPs
    2. 2.2 Split locus and balanced split locus
    3. 2.3 Balanced property of the Finsler cut locus
  9. 3 Local structure of cut and singular loci up to codimension 3
    1. 3.1 Statements of results
      1. 3.1.1 Results
      2. 3.1.2 Examples
      3. 3.1.3 Relation to previous results in the literature
    2. 3.2 Conjugate points in a balanced split locus
    3. 3.3 Structure up to codimension 3
  10. 4 Balanced split sets and Hamilton-Jacobi equations
    1. 4.1 Introduction
      1. 4.1.1 Outline
    2. 4.2 Statement of results.
      1. 4.2.1 Results
      2. 4.2.2 Examples
    3. 4.3 Preliminaries
      1. 4.3.1 Lagrangian submanifolds of $T^{\ast}M$
      2. 4.3.2 A useful lemma
      3. 4.3.3 Some generalities on HJ equations.
    4. 4.4 $\rho_{S}$ is Lipschitz
    5. 4.5 Proof of the main theorems.
      1. 4.5.1 Proof of Theorem 4.2.1.
      2. 4.5.2 Proof of Theorem 4.2.2.
      3. 4.5.3 Proof of Theorem 4.2.4.
    6. 4.6 Proof that $\partial T=0$
      1. 4.6.1 Conjugate points of order $1$.
      2. 4.6.2 Structure of S near cleave and crossing points
      3. 4.6.3 Conclusion
  11. 5 A new proof of the Ambrose conjecture for generic $3$-manifolds
    1. 5.1 Introduction
      1. 5.1.1 Cartan lemma
      2. 5.1.2 Ambrose Conjecture
      3. 5.1.3 Existing results
    2. 5.2 Notation and preliminaries
      1. 5.2.1 The approach of James Hebda using tree-formed curves
      2. 5.2.2 Difficulties to extend the proof to dimension higher than $2$
    3. 5.3 Generic exponential maps
      1. 5.3.1 Generalized phase functions
      2. 5.3.2 The singularities of a generic exponential map
    4. 5.4 Proof of the conjecture for generic $3$-manifold
      1. 5.4.1 Main idea
      2. 5.4.2 Synthesis
      3. 5.4.3 Proof that $A_{3}(I)$ first conjugate points are unequivocal
      4. 5.4.4 Conjugate flow
      5. 5.4.5 CDCs in adapted coordinates near $A_{3}$ points
      6. 5.4.6 $A_{3}$ joins
      7. 5.4.7 Avoiding some obstacles
      8. 5.4.8 Linking curves
      9. 5.4.9 Existence of linking curves for easy manifolds from a point
      10. 5.4.10 Existence of positive transient pairs in easy manifolds
      11. 5.4.11 Proof that $\pi_{1}$ and $\pi_{2}$ are covering maps
      12. 5.4.12 Proof of 5.4.21
      13. 5.4.13 Proof for a $3$-manifold with a generic metric
      14. 5.4.14 CDCs in adapted coordinates
      15. 5.4.15 Proof of theorem 5.4.32
  12. 6 Further questions
    1. 6.1 Order $k$ conjugate cut points
    2. 6.2 HJBVP and balanced split loci
    3. 6.3 Poincaré conjecture
    4. 6.4 Magician’s hat
    5. 6.5 Proof for a 3-manifold with an arbitrary metric
      1. 6.5.1 Main idea
      2. 6.5.2 Proof of the Ambrose conjecture modulo the IJK conjecture
      3. 6.5.3 Terminal points for the conjugate descending flow
  13. 7 Conclusiones

## Introducción

En esta tesis estudiamos las singularidades de la aplicación exponencial en
variedades Riemannianas y Finslerianas, y el objeto conocido en inglés como
_cut locus_ , _ridge_ , _medial axis_ o _skeleton_ , de los cuales sólo el
último término suele traducirse al castellano. En primer lugar mejoramos los
resultados existentes sobre las singularidades de la aplicación exponencial y
la estructura del cut locus, y después aplicamos estos resultados a los
problemas de frontera para ecuaciones de Hamilton-Jacobi y a la conjectura de
Ambrose. El cut locus es un objeto de interés para muchas disciplinas:
geometría diferencial, teoría de control óptimo, teoría de transporte óptimo,
procesamiento de imágenes, estadística y una herramienta útil en algunas
demostraciones de resultados en otras disciplinas en las que el cut locus en
sí no es un objeto de interés directo. Durante la primera fase recogimos
resultados sobre la estructura del cut locus provenientes de muchas de estas
disciplinas, encontrando resultados duplicados, y mucho desconocimiento en
cada área del trabajo que sobre este objeto se hacía desde las otras
disciplinas. Cuando aportamos nuestros propios resultados, tuvimos que elegir
una notación que no podía ser compatible con toda la literatura existente.
Nuestros resultados sobre estructura en los capítulos 3 y 4 generalizan
resultados bien conocidos y demostrados muchas veces de forma independiente,
que describen la estructura del cut locus excepto por un conjunto de
codimensión $2$, lo que es útil para muchas aplicaciones, pero no para todas,
aumentando el conocimiento del cut locus hasta codimensión $3$. Estos
resultados de estructura son esenciales para nuestras aportaciones a la teoría
de Problemas de Frontera para Ecuaciones de Hamilton-Jacobi en el capítulo 4,
donde conectamos la noción de solución de viscosidad con la solución clásica
por características caracterizando el lugar singular de la primera como un cut
locus, o como un _balanced split locus_ , noción que identificamos en este
trabajo aunque estaba implícito en trabajos previos. Creemos que los
resultados sobre las singularidades de la aplicación exponencial del capítulo
3 podrían ser útiles para extender la demostración de la conjetura de Ambrose
que aportamos a todas las métricas riemannianas. En el capítulo 5, damos una
demostración nueva de la conjetura de Ambrose que cubre un conjunto genérico
de variedades riemannianas, pero en el capítulo 6, pergeñamos una estrategia
que podría servir para dar una demostración más general que usa de forma
esencial los resultados de estructura mencionados. El resto de esta
disertación doctoral será en inglés para ser útil a un público más amplio,
esperamos que este hecho no suponga un impedimento al lector interesado.

## Acknowledgements

Yanyan Li introduced me to the Ambrose conjecture and sparked my interest in
the conjecture during my stay at Rutgers University. He is a great teacher,
mathematician, and human being. During the preparation of the papers [AG1] and
[AG2], I had many helpful conversations with young and senior mathematicians.
The list includes Biao Yin, Luc Nguyen, Juan Carlos Álvarez Paiva, Ireneo
Peral, Yanyan Li and Marco Fontelos. Finally, the referee of [AG2] was very
helpful. Whoever that was, please receive my warmest regards. I have also
talked with many people about the Ambrose conjecture. The list includes Juan
Carlos Álvarez Paiva, Paolo Piccione, Herman Gluck. Other resources were more
impersonal but equally useful. Neil Strickland and Ben Wieland answered a
question I posted in the algebraic topology list ALGTOP-L. The Wikipedia
helped save a lot of time by providing quick answers to many simple questions.
The site mathoverflow already contained answers to a few questions before we
could even word them correctly. Some anonymous mathematicians scanned,
uploaded and shared a big mathematical library. They helped a lot, and they
have my respect and my full support. I hold even more respect for all the
mathematicians that released their work directly to the public. The wonderful
book of Allen Hatcher was particularly helpful. You even can find one picture
from his book in this thesis (with permission of the author). A special place
goes to the Sage community, an open source mathematical software that I’ve
used mainly for teaching, but also to do computations and explore some
hypothesis related to this thesis. I’m proud to have been part of that
community for several years. Luis Guijarro was always helpful and respectful
as a thesis advisor. Our friendship has grown during these years and will
survive this work. His patience knows no bounds. During most of the years I
worked on this thesis, I was lucky to work on the Mathematics Department of
the Universidad Autónoma de Madrid. The different universities and research
centers at Madrid make it a great place to stay tuned with the latests
advances in mathematics, but it is the warmth and fellowship of the people in
the department that made those years so pleasant. In particular, I want to
recall my office mates Pedro Caro and Carlos Vinuesa, with whom I had many
laughs and interesting mathematical conversations, and Daniel Ortega, who
taught me by example that is more satisfying to be useful to the department
than to improve the cv. Daniel is also the latex guru of the department, and
is responsible for fixing all the badboxes and other stylistic errors in this
thesis. Besides my positions in the department, I was partially supported
during the preparation of this work by grants MTM2007-61982 and MTM2008-02686
of the MEC and the MCINN respectively. At the end of this work, I was
supported by the Instituto Nacional de Empleo.

###### Agradecimientos

Let me switch now to spanish: the rest is more personal. La casualidad quiso
que mi amigo Daniel estuviera en la UAM preparando su tesis cuando entré como
profesor ayudante. Dani me abrió muchas puertas, y durante estos años, al
igual que siempre, fue una persona dispuesta a escuchar cualquier problema en
cualquier área y a proponer soluciones creativas. María y Clara han sido mis
compañeras durante toda mi vida matemática. Con ellas he compartido
matemáticas y muchas otras cosas. Creo que me han hecho crecer como persona de
un modo tan profundo que listar sus contribuciones a esta tesis en particular
sería frívolo. De no haberlas conocido, otra persona habría escrito este
trabajo. Clara y mis dos hijos César y Héctor son mi pasión y fuerza vital.
Subido a hombros de estos gigantes, afronto el futuro con alegría, incluso con
optimismo, aún intuyendo algunos de los momentos difíciles que nos esperan,
confiado de que podrán superar todas las dificultades. Pero quiero dedicar
esta tesis a mis padres. Desde niño me he sentido siempre orgulloso de mi
madre. Incluso ahora, siento que tengo mucho que aprender de ella para poder
ser el padre que mis hijos merecen. Mi padre es ahora el abuelo de mis hijos,
y es un orgullo haber contribuido a elevarle a este status. En un hombre con
muchas virtudes, su estilo y saber hacer como abuelo se elevan sobre todas las
demás. De todo el legado de mis padres, sin duda su impronta sobre sus hijos y
sus nietos será la más importante y duradera. Su gran humanidad me acompaña y
me da fuerza en los momentos fáciles y difíciles.

## Introduction

The goal of this thesis is to study the singularities of the exponential map
of _Riemannian and Finsler manifolds_ (a concept related to _caustics_ and
_catastrophes_), and the object known as the cut locus (aka _ridge_ , _medial
axis_ or _skeleton_ , with applications to differential geometry, control
theory, statistics, image processing…), to improve existing results about its
structure, to look at it in new ways, and to derive applications to the
Ambrose conjecture and the Hamilton-Jacobi equations.

###  I. Relation between Hamilton-Jacobi equations and Finsler Geometry

Boundary Value Problems of Hamilton-Jacobi (HJBVP) are intimately relationed
to Finsler Geometry. In such problems, we look for an unknown function
$u:M\rightarrow\mathbb{R}$ satisfying the following equations:

$\displaystyle H(p,du(p))\;=$ $\displaystyle 1\quad$ $\displaystyle p\in M$
$\displaystyle u(p)\;=$ $\displaystyle g(p)$ $\displaystyle p\in\partial M$

where the first equation is a _non-linear first order partial differential
equation_ and the second equation prescribes the _boundary values_ for $u$. We
ask for the following conditions:

  * •

$M$ is a smooth compact manifold of dimension $n$ with boundary

  * •

$H:T^{\ast}M\rightarrow RR$ is a smooth function defined on the _cotangent
space_ to $M$, $H^{-1}(1)\cap T^{\ast}_{p}M$ strictly convex for every $p$

  * •

$g:\partial M\rightarrow\mathbb{R}$ smooth

Furthermore, the _boundary data_ $g$ and the _equation coefficients_ $H$ must
satisfy a compatibility condition:

$|g(y)-g(z)|<d(y,z)\qquad\forall y,z\in\partial M$

where $d$ is the distance on $M$ induced by the following _Finsler metric_ :

$\varphi_{p}(v)=\sup\left\\{\left\langle
v,\alpha\right\rangle_{p}\,:\,\alpha\in
T^{\ast}_{p}M,\,H(p,\alpha)=1\right\\}$

The above definition gives a norm in every tangent space $T_{p}M$. Indeed, $H$
can be redefined so that $H$ is positively homogeneous of order $1$:
$H(p,\lambda\alpha)=\lambda H(p,\alpha)$ for $\lambda>0$, and the HJBVP is the
same. Then, $H$ is a norm at every cotangent space and $\varphi$ is the dual
norm in the tangent space. A classical solution to these equations has been
known for a long time, and it admits a geometrical interpretation in terms of
Finsler geometry. First, using the definition of dual form in Finsler geometry
(see 1.1.6), we define the _characteristic vector field_ at points
$p\in\partial M$:

$\displaystyle\varphi_{p}(X_{p})=1$ $\displaystyle\widehat{X_{p}}|_{T(\partial
M)}=dg$ $\displaystyle X_{p}\text{ points inwards}$

The (projected) characteristic curves are the geodesics $M$ with initial point
$z\in\partial M$ and initial speed given by the characteristic vector field. A
local smooth solution $u$ to the HJBVP can be computed near $\partial M$
following characteristic curves:

######  Definition.

Let $U$ be a neighborhood of $\partial M$ such that every point $q\in U$
belongs to a unique (projected) characteristic contained in $U$ and starting
at a point $p\in\partial M$ (the point $p$ is often called the _footpoint_ of
$q$). The solution by characteristics $u:U\rightarrow\mathbb{R}$ is defined as
follows: if $\gamma:[0,t]\rightarrow M$ is the unique (projected)
characteristic from a point $p\in\partial M$ to $q=\gamma(t)$ that does not
intersect $Sing$, then

$u(q)=g(p)+t$

In this way, the classical solution can be defined in a neighborhood of
$\partial M$, but not in all of $M$. A different notion of solution appeared
later (see [L]). The solution (in the _viscosity_ sense) to the above HJBVP is
given by the Lax-Oleinik formula:

$u(p)=\inf_{q\in\partial M}\left\\{d(p,q)+g(q)\right\\}$

where $d$ is again the distance function induced from the Finsler metric. We
defer the definition to 1.4.2, because the actual definition of the viscosity
solution plays no role in this thesis. All we need to know is that the
viscosity solution is given by the above formula. Thus, when $g=0$, the
solution to the equations is the distance to the boundary. In theorem 2.1.6,
we prove that when $g\neq 0$, the viscosity solution is also a distance
function, but to the boundary of a larger manifold $\tilde{M}\supset M$.

###  II. Singularities of the exponential map

The _exponential map_ from a point or submanifold in a Finsler manifold is
defined in the same way as that of Riemannian manifolds and has similar
properties. Let $M$ be a smooth Finsler manifold, $p$ a point in $M$ and $v\in
T_{p}M$ a tangent vector to $M$ at $p$. Then the exponential of $v$ is the
point $\exp_{p}(v)=\gamma(1)\in M$, for the unique geodesic $\gamma$ that
starts at $p$ and has initial speed vector $v$. The exponential map from $p$
is a diffeomorphism from a small ball near the origin, but it can develop
singularities as we move far away from the origin. The exponential map from a
submanifold $L\subset M$ is defined for (some) vectors of the normal bundle of
$L$ in $M$: let $p$ be a point in $L$ and $v\in T_{p}M$ a vector orthogonal to
the subspace $T_{p}L$, the exponential of $v$ is the point
$\exp_{L}(v)=\gamma(1)\in M$, for the unique geodesic $\gamma$ that starts at
$p$ and has initial speed vector $v$. If the submanifold is the boundary
$\partial M$ of a closed manifold $M$, the definition is the same, but the
exponential is only defined in the _inner normal bundle_. This time, the
exponential map is a diffeomorphism from a tubular neighborhood of the zero
section of the normal bundle of $L$ into a tubular neighborhood of $L$ in $M$
and again, it can develop singularities and self-intersections if we consider
larger vectors. The singularities, however, are not those of an arbitrary
smooth map between $n$-dimensional manifolds. Let us restrict for a moment to
the exponential map from a point $p$ in a manifold $M$ without boundary. For a
point $x\in T_{p}M$ where $\exp_{p}$ is singular, the _order of conjugacy_ of
$x$ is the corank of the linear map $d_{x}\exp_{p}$. Along a radial line in
$T_{p}M$, the singularities cannot cluster: if we add the orders of all the
singularities along a radial line, in a small neighborhood of a point $x\in
T_{p}M$ of order $k$, the number is always $k$. It makes sense, thus, to talk
about _the $k$-th conjugate point in the direction $x$_, for a point $x$ in
the unit ball in $T_{p}M$. This is the point $t_{0}\cdot x$, for $t_{0}>0$,
such that $t_{0}\cdot x$ is a conjugate point of order $j$ and such that the
sum of the orders of all the singularities along the radial segment
$t\rightarrow t\cdot x$, for $t<t_{0}$ is an integer between $k-j$ and $k-1$.
This allows us to define the function $\lambda_{k}$ that maps $x\in
B_{1}(T_{p}M)$ to the parameter $t=\lambda_{k}(x)$ such that $t\cdot x$ is the
the $k$-th conjugate point in the directon $x$. It follows from the above that
$\lambda_{k}$ is continuous. In [IT98], J. I. Itoh and M. Tanaka proved that
for Riemannian manifolds, all the functions $\lambda_{k}$ are locally
Lipschitz continuous. In theorem 4.2.5, we prove that all $\lambda_{k}$ are
locally Lipschitz continuous in Finsler manifolds. M. Castelpietra and L.
Rifford were working on this result simultaneously, and shortly after our
proof appeared, they gave a proof that $\lambda_{1}$ is locally semiconcave, a
stronger property that Lipschitz. This property does not hold for
$\lambda_{k}$, for $k>1$. The three proofs are different.

###  III. Structure of the cut locus

Let $M$ be a Riemannian or Finsler manifold, and let $S$ be a smooth
submanifold of any dimension. $S$ can also be a point, and we also consider
$S=\partial M$. It is easy to see that this latter case contains the others,
substracting a tubular neighborhood of the submanifold $S$. The cut locus
$Cut_{S}$ of $S$ in $M$ can be defined in several equivalent ways:

---
Figure III.1. Two open sets in $\mathbb{R}^{2}$ with their respective cut loci
(with respect to the boundary). The second set has analytic boundary, and its
cut locus is a subanalytic set.

  * •

Every unit speed geodesic $\gamma$ with initial point in $S$ and initial speed
orthogonal to $S$ will minimize the distance between $\gamma(0)$ and
$\gamma(t)$, for small $t$. Define
$t_{cut}=\sup\\{t:d(\gamma(0),\gamma(t))=t\\}$. The cut locus is the set of
all points $\gamma(t_{cut}(\gamma))$ for all geodesics $\gamma$ starting at
$S$ with initial speed orthogonal to $S$.

  * •

For a point $p\in M$, let $Q_{p}$ be the set of points $q\in S$ such that
$d(p,q)=d(p,S)$. Then $Cut_{S}$ is the closure of the set of points such that
$Q_{p}$ has more than one element.

  * •

The distance function to $S$ is singular exactly when $Q_{p}$ has more than
one element, so we can also define $Cut_{S}$ as the closure of the set where
the distance function to $S$ is singular.

  * •

$Cut_{S}$ is also the set of points such that either $Q_{p}$ has more than one
element, or $Q_{p}=\\{q\\}$, and $q$ is conjugate to $p$ along one geodesic
that minimizes the distance between them.

These sets have been studied by mathematicians from many different fields:

  * •

$Cut_{\partial M}$ is a deformation retract of $M$ (and $Cut_{S}$ is a
deformation retract of $M\setminus S$ ) (obvious).

  * •

$Cut_{S}$ is the union of a ($n-1$)-dimensional smooth manifold consisting of
points with two minimizing non-conjugate geodesics and a set of Hausdorff
dimension at most $n-2$ (Hebda83, Itoh-Tanaka98, Barden-Le97, Mantegazza-
Menucci03 for the riemannian case).

  * •

$Cut_{S}$ is stratified by the dimension of the subdifferential of the
function _distance to $S$_: $\partial d_{S}$ ( Alberti-Ambrosio-Cannarsa-
Etcetera92-94).

  * •

It has finite Hausdorff measure $\mathcal{H}^{n-1}$ (Itoh-Tanaka00 for the
riemannian case, Li-Nirenberg05 and Castelpietra-Rifford10 for Finsler
manifolds).

  * •

If all the data is analytic, $Cut_{S}$ is a stratified _subanalytic set_ (
Buchner77).

  * •

If we add a generic perturbation to $H$ or $M$, $Cut_{S}$ becomes a stratified
smooth manifold (Buchner78).

Despite these facts, cut loci can be non-triangulable, even for surfaces of
revolution (see [GS]). Also, their combinatorial topology can be complicated:
even though $Cut_{S}$ is homotopic to $M\setminus S$, there are metrics in the
$3$-dimensional sphere whose cut locus is a simplicial complex equivalent to
the house with two rooms (see figure III.2).

Figure III.2. The house with two rooms

We have improved the previous knowledge about _cut loci_ and the _singular
set_ of solutions to static Hamilton-Jacobi equations in the following ways:

Figure III.3. Different types of points in the cut locus, except for a set of
small Haussdorff dimension

  * •

In theorem 2.1.8, we show that the _singular set_ of a solution to a Hamilton-
Jacobi BVP is the cut locus of a Finsler manifold. This was only known when
the boundary data was identically zero.

  * •

In the same chapter, in subsection 2.3, we also show that cut loci are what we
call _balanced split loci_. In general, there are many balanced split loci
besides the cut locus, but there is only one such set on a simply connected
manifold with connected boundary.

  * •

In theorem 3.1.2, we prove a local structure theorem for balanced split loci.
The theorem states that the cut locus consists only of _cleave points_ , _edge
points_ , _degenerate cleave points_ , _crossing points_ , and a remainder, of
Hausdorff codimension at least 3 (see figure III.3).

  * •

In chapter 4, we provide more detailed descriptions of a balanced split loci
near the different types of points listed above (see 4.6.2, 4.6.3, 4.6.5,
4.6.6 and 4.6.7).

We believe that our description of the cut locus can be useful in other
contexts. For instance, brownian motion on manifolds is often studied on the
complement of the cut locus from a point, and then the results have to be
adapted to take care of the situation when the brownian motion hits the cut
locus. As brownian motion almost never hits a set with null
$\mathcal{H}^{n-2}$ measure (but will almost surely hit any set with positive
$\mathcal{H}^{n-2}$ measure), we think our result can be useful in that field.

###  IV. Characterization of the cut locus and the balanced split loci

As we mentioned in the previous section all cut loci, and thus the singular
set of HJBVP, are _balanced split loci_. In chapter 4, we study and classify
all possible balanced split loci. The following list is a summary of theorems
4.2.1, 4.2.2 and 4.2.4:  $M$ is simply connected $\rightarrow$ The singular
set is the unique _and_ $\partial M$ connected balanced split locus $M$ is
simply connected, $\rightarrow$ We can add a different constant $\partial M$
is __not__ connected to $g$ at each component of $\partial M$ and get
different balanced split loci (see fig IV.4) General case $\rightarrow$
Balanced split loci are parametrized by a neighborhood of $0$ in
$H_{n-1}(M,\mathbb{R})$ (see fig IV.5)

Figure IV.4. A balanced split locus when $\partial M$ consists of two
concentric spheres Figure IV.5. A balanced split locus of a non-simply-
connected manifold (a torus with a disc removed)

###  V. The Ambrose conjecture

Let ($M_{1},g_{1}$) and ($M_{2},g_{2}$) be two Riemannian manifolds _of the
same dimension_ , with selected points $p_{1}\in M_{1}$ and $p_{2}\in M_{2}$.
We’ll speak about the pointed manifolds ($M_{1},p_{1}$) and ($M_{2},p_{2}$).
Any linear map $L:T_{p_{1}}M_{1}\rightarrow T_{p_{2}}M_{2}$ induces the map
$\varphi=\exp_{2}\circ L\circ(\exp_{1}|_{O_{1}})^{-1}$, defined in any domain
$O_{1}\subset T_{p_{1}}M_{1}$ such that $e_{1}|_{O_{1}}$ is injective (for
example, if $O_{1}$ is a normal neighborhood of $p_{1}$).

Figure V.6. The map $\varphi$ induced by the linear map $L$

This idea was introduced by E. Cartan [C], who proved that under some (strong)
hypothesis on the curvature of $M_{1}$ and $M_{2}$, this map is a local or
even a global isometry. We prefer to rephrase it in the following terms:

######  Definition.

Let $(M_{1},p_{1})$ and $(M_{2},p_{2})$ be complete Riemannian manifolds of
the same dimension with base points, and $L:T_{p_{1}}M_{1}\rightarrow
T_{p_{2}}M_{2}$ a linear isometry. Let $\gamma_{1}$ and $\gamma_{2}$ be the
geodesics defined in the interval $[0,1]$, with $\gamma_{1}$ starting at
$p_{1}$ with initial speed vector $x\in T_{p_{1}}M_{1}$ and $\gamma_{2}$
starting at $p_{2}$ with initial speed $L(x)$. For any three vectors
$v_{1},v_{2},v_{3}$ in $T_{p_{1}}M_{1}$, define:

  * $\bullet$

$R_{1}(x,v_{1},v_{2},v_{3})$ is the vector of $T_{p_{1}}M_{1}$ obtained by
performing parallel transport of $v_{1},v_{2},v_{3}$ along $\gamma_{1}$,
computing the Riemann curvature tensor at the point $\gamma_{1}(1)\in M_{1}$
acting on those vectors, and then performing parallel transport backwards up
to the point $p_{1}$.

  * $\bullet$

$R_{2}(x,v_{1},v_{2},v_{3})$ is the vector of $T_{p_{1}}M_{1}$ obtained by
performing parallel transport of $L(v_{1}),L(v_{2}),L(v_{3})$ along
$\gamma_{2}$, computing the Riemann curvature tensor at the point
$\gamma_{2}(1)\in M_{2}$ acting on those vectors, then performing parallel
transport backwards up to the point $p_{2}$, and finally applying $L^{-1}$ to
get a vector in $T_{p_{1}}M_{1}$.

If $R_{1}(x,v_{1},v_{2},v_{3})=R_{2}(x,v_{1},v_{2},v_{3})\forall
v_{1},v_{2},v_{3}\in T_{p_{1}}M_{1}$ for any two geodesics $\gamma_{1}$ and
$\gamma_{2}$ as above, we say that _the curvature tensors of $(M_{1},p_{1})$
and $(M_{2},p_{2})$ are $L$-related_.

The usual way to express that the curvature tensors of $(M_{1},p_{1})$ and
$(M_{2},p_{2})$ are $L$-related is to say that the _parallel traslation of
curvature along radial geodesics_ of $(M_{1},p_{1})$ and $(M_{2},p_{2})$
coincides. We say $(M_{1},p_{1})$ and $(M_{2},p_{2})$ are $L$-related iff they
have the same dimension and, whenever $\exp_{1}|_{O_{1}}$ is injective for
some domain $O_{1}\subset T_{p_{1}}M_{1}$, then the map $\varphi=\exp_{2}\circ
L\circ(\exp_{1}|_{O_{1}})^{-1}$ is an isometric inmersion. Cartan’s theorem
states that if the curvature tensors of $(\\!M_{1},p_{1})$ and $(M_{2},p_{2})$
are $L$-related, then $(\\!M_{1},p_{1}\\!)$ and $(\\!M_{2},p_{2}\\!)$ are
$L$-related. (see lemma 1.35 of [CE]). In 1956 (see [A]), W. Ambrose proved a
global version of the above theorem, but with stronger hypothesis. A _broken
geodesic_ is the concatenation of a finite amount of geodesic segments. The
theorem of Ambrose states that if the parallel traslation of curvature along
_broken geodesics_ on $M_{1}$ and $M_{2}$ coincide, then there is a global
isometry $\varphi:M_{1}\rightarrow M_{2}$ whose differential at $p_{1}$ is
$L$. It is simple to prove that $\varphi$ can be constructed as above. It is
enough if the hypothesis holds for broken geodesics with only one “ _elbow_ ”
(the reader can find more details in [CE]). However, he conjectured that the
same hypothesis of the Cartan’s lemma should suffice, except for the obvious
counterexamples of covering spaces: The Ambrose conjecture states that if the
curvature tensors of $(M_{1},p_{1})$ and $(M_{2},p_{2})$ are $L$-related, and
$M_{1}$ and $M_{2}$ are _simply connected_ , there is a global isometry
$\psi:M_{1}\rightarrow M_{2}$ such that $\psi\circ\exp_{1}=\exp_{2}\circ L$.
Ambrose himself was able to prove the conjecture if all the data is analytic.
In [Hi], in 1959, the conjecture was generalized to parallel transport for
affine connections, and in [BH], in 1987, to Cartan connections. Also in 1987,
in the paper [H87], James Hebda proved that the conjecture was true for
surfaces that satisfy a certain regularity hypothesis, that he was able to
prove true in 1994 in [H94]. J.I. Itoh also proved the regularity hypothesis
independently in [I]. The latest advance came in 2010, after we had started
our research on the Ambrose conjecture, when James Hebda proved in [H10] that
the conjecture holds if $M_{1}$ is a _heterogeneous manifold_. Such manifolds
are generic. In 5.1.6 we provide a new proof that works for surfaces and for a
generic class of manifolds in dimension $3$. James Hebda’s proof in [H10] is
shorter than ours and works for any dimension. However, his proof does not
extend to arbitrary metrics and we think that our proof might, even though we
have been unable to complete all the details to this day. Indeed, the proof
presented here extends to some manifolds that are not covered by the result of
J. Hebda, as this is truly a different approach. In chapter 6 we also provide
some hints on how our proof might extend to a $3$-dimensional manifold with an
arbitrary metric. We remark that in both our proof and James Hebda’s, it is
only used that $(M_{1},p_{1})$ and $(M_{2},p_{2})$ are $L$-related, and there
is no need to use the original hypothesis that the curvarture tensors are
$L$-related.

###  VI. Summary of results

  * •

Reduction of a HJBVP with $g\neq 0$ to a HJBVP with $g=0$: theorem 2.1.6,
published in [AG1].

  * •

Proof that all $\lambda_{k}$ are locally Lipschitz continuous in Finsler
manifolds: theorem 4.2.5, published in [AG1].

  * •

Proof that the _singular set_ of a solution to a Hamilton-Jacobi BVP is the
cut locus of a different manifold: theorem 2.1.8, published in [AG1].

  * •

Proof that cut loci in Finsler manifolds are _balanced split loci_ : section
2.3, published in [AG1].

  * •

Local structure theorem for balanced split loci: theorem 3.1.2, published in
[AG1].

  * •

Detailed descriptions of a balanced split loci near the different types of
points, except for a set of Hausdorff codimension 3: 4.6.2, 4.6.3, 4.6.5,
4.6.6 and 4.6.7, published in [AG2].

  * •

Characterization of the cut locus and the balanced split loci: theorems 4.2.1,
4.2.2 and 4.2.4, published in [AG2].

  * •

Proof of the Ambrose conjecture for a metric in $\mathcal{G}_{M}$: theorem
5.1.6, publication pending.

  * •

A whole selection of fresh conjectures for the new generations in chapter 6…

##  Chapter 1 Preliminaries

#### Notation

We fix the following notation for the rest of the thesis:

  * •

A $C^{\infty}$ manifold $M$ with boundary $\partial M$.

  * •

A Finsler metric $\varphi$ on $M$.

  * •

The distance $d$ induced on $M$ by $\varphi$, and the “distance to the
boundary” $d_{\partial M}(p)=\inf_{q\in\partial M}d(q,p)$.

  * •

The geodesic vector field $\rho$ in $TM$.

  * •

The time-$t$ flow of $\rho$: $\Phi_{t}:TM\rightarrow TM$. For a non-complete
manifold, such as a manifold with boundary, these maps are not defined in all
of $TM$.

  * •

A smooth map $\Gamma:\partial M\rightarrow TM$ that is a section of the
projection map $\pi:TM\rightarrow M$ of the tangent to $M$, and such that
$\Gamma(p)$ points to the inside of $M$ for every $p\in\partial M$.

###  1.1. A little background

####  1.1.1. Approximate Tangent Cone

######  Definition 1.1.1.

For a pair of points $p,q\in M$ such that $q$ belongs to a convex neighborhood
of $p$, we define, following [IT00],

(1.1.1) $v_{p}(q)=\dot{\gamma}(0)$

as the speed at $0$ of the unique unit speed minimizing geodesic $\gamma$ from
$p$ to $q$.

######  Definition 1.1.2.

The _approximate tangent cone_ to a subset $E\subset M$ at $p$ is:

$T(E,p)=\left\\{rv:\;v=\lim\frac{v_{p}(p_{n})}{|v_{p}(p_{n})|},\exists\\{p_{n}\\}\subset
E,p_{n}\rightarrow p,r>0\right\\}$

and the _approximate tangent space_ $Tan(E,p)$ to $E$ at $p$ is the vector
space generated by $T(E,p)$.

We remark that the definition is independent of the Finsler metric, despite
its apparent dependence on the vectors $v_{p}(p_{n})$.

####  1.1.2. Subdifferentials of semiconcave functions

Concave (or convex) functions $u$ may not be differentiable, but they are
differentiable almost everywhere. This allows for a simple definition of the
subdifferential of a convex or semiconvex function (see [CS] for different
definitions):

######  Definition 1.1.3.

The _subdifferential_ $\partial u(p)$ of a concave function $u$ at $p$ can be
defined as the convex hull of all the one forms that are limits of
differentials of $u$ at points $p_{n}$ where $u$ is differentiable.

######  Definition 1.1.4.

A function $u:S\rightarrow\mathbb{R}$ is semiconcave if there exists a
nondecreasing upper semicontinuous function
$\omega:\mathbb{R}_{+}\rightarrow\mathbb{R}+$ such that
$\lim\limits_{\rho\rightarrow 0}\omega(\rho)=0$ and, for any
$x,y\in\mathbb{R}$ and $\lambda\in[0,1]$:

$\lambda u(x)+(1-\lambda)u(y)-u(\lambda
u+(1-\lambda)y)\leq\lambda(1-\lambda)|x-y|\omega(|x-y|)$

The function $\omega$ is called the _modulus of semiconcavity_.

The concave functions are those for which $\omega$ is zero. The functions with
a _linear modulus_ of semiconcavity are exactly those that can be written as
the sum of a concave and a smooth function. It turns out that _viscosity_
solutions to HJBVP (to be defined later), and distance functions to the
boundary in Finsler geometry, are semiconcave functions. Those functions share
many of the regularity results of concave functions. For example, they are
differentiable almost everywhere. Indeed, this statement can be refined: let
$u:S\rightarrow\mathbb{R}$ be a semiconcave function and define the sets

$\Sigma_{k}=\\{x\in S:\dim(\partial u(p))\geq k\\}$

Then $\Sigma_{k}$ is countably $\mathcal{H}^{n-k}$ rectifiable: it is
contained in the union of countably many $C^{1}$ hypersurfaces of dimension
$n-k$, plus a $\mathcal{H}^{n-k}$-negligible set. Check [CS] and [AAC] for
more details.

####  1.1.3. Duality in Finsler Geometry

######  Definition 1.1.5.

The orthogonal hyperplane to a vector $v\in T_{p}M$ is the hyperplane tangent
at $v$ to the level set

$\left\\{v^{\prime}\in T_{p}M:\varphi(v^{\prime})=\varphi(v)\right\\}$

The orthogonal distribution to a vector field is defined pointwise.

Remark. There are two unit vectors with a given hyperplane as orthogonal
hyperplane. The first need not to be the opposite of the second unless $H$ is
symmetric ($H(-v)=H(v)$). We thus define two unit normal vectors to a
hypersurface (the _inner_ normal and _outer_ normal).

######  Definition 1.1.6.

The dual one form to a vector $v\in T_{p}M$ with respect to a Finsler metric
$\varphi$ is the unique one form $\omega\in T^{\ast}_{p}M$ such that
$\omega(v)=\varphi(v)^{2}$ and $\omega|_{H}=0$, where $H$ is the orthogonal
hyperplane to $v$. For a vector field $X$, the dual differential one-form is
obtained by applying the above construction at every point. We will often use
the notation $\widehat{X}$ for the dual one-form to the vector field $X$.

Remark. In Riemannian geometry, a different scaling is often used:
$\omega(v)=1$ instead of $\omega(v)=\varphi(v)^{2}$. We have chosen this
definition because it makes the duality map $v\rightarrow\omega$ continuous.
Remark. In coordinates, the dual one form $w$ to the vector $v$ is given by:

$w_{j}=\varphi(v)\frac{\partial\varphi}{\partial v^{j}}(p,v)$

Actually $\varphi$ is 1-homogeneous, so Euler’s identity yields:

$w_{j}v^{j}=\varphi(v)\frac{\partial\varphi}{\partial
v^{j}}(p,v)v^{j}=\varphi(v)^{2}$

and, for a curve $\gamma(-\varepsilon,\varepsilon)\rightarrow T_{p}M$ such
that $\gamma(0)=v$, $\varphi(\gamma(t))=\varphi(v)$ and
$\gamma^{\prime}(0)=z$,

$\varphi(v)w_{j}z^{j}=\varphi(v)^{2}\frac{\partial}{\partial
t}|_{t=0}\varphi(\gamma(t))=0$

Remark. The hypothesis on $\varphi$ imply that the orthogonal form to a vector
is unique, and the correspondence between a vector and its dual one form is
one to one, but it is only linear for riemannian metrics.

###  1.2. Exponential maps of Finsler Manifolds

######  Definition 1.2.1.

Let $D(\Phi_{t})$ be the domain of the time-$t$ flow of the geodesic vector
field in $TM$. We introduce the sets $V$ and $W$:

(1.2.1) $V=\left\\{(t,p):t\geq 0,p\in\partial M,\Gamma(p)\in
D(\Phi_{t})\right\\}\subset\mathbb{R}\times\partial M$ (1.2.2)
$W=\left\\{\Phi_{t}(\Gamma(p)):t\geq 0,p\in\partial M,\Gamma(p)\in
D(\Phi_{t})\right\\}\subset TM$

$V$ and $W$ are diffeomorphic through the map $G(t,p)=\Phi_{t}(\Gamma(p))$. We
define the exponential map $F$ associated to $(M,\Gamma)$ as $F=\pi\circ
G:V\rightarrow M$. The interior of $W$ is locally invariant under $\Phi_{t}$.
This is equivalent to saying that $\rho$ is tangent to $W$. The radial vector
$r=\frac{\partial}{\partial t}$ is mapped to $\rho$ by $G_{\ast}$.

######  Remark 1.2.2.

The map $G$ is injective in its domain; its inverse can be computed walking a
geodesic backwards until we hit the boundary for the first time. In other
words, $W$ is a smooth manifold and $G$ is a diffeomorphism from
$V\subset\mathbb{R}\times\partial M$.

######  Remark 1.2.3.

The map $G$ can also be written $G(t,p)=(F(p),dF_{p}(r))$, as follows from the
geodesic equations.

In the particular case when $\Gamma(p)$ is the inner unit normal vector to
$\partial M$, this is the standard “exponential map from $\partial M$”. This,
in turn, includes the exponential map from a point $p$ in a manifold without
boundary, in the following way: Let $p$ be a point in a manifold without
boundary, and remove a ball $B$ of small radius around $p$. The result is a
manifold with boundary $\partial M=\partial B$, and the exponential map from
$\partial B\subset M\setminus B$ coincides with the exponential map from $p\in
M$. The same trick works for the exponential from a submanifold of any
codimension inside a manifold without boundary, removing a tubular
neighborhood around the submanifold. The motivation for working in the above
setting is that it also works for the Hamilton-Jacobi BVP with non-trivial
boundary data, and it does not make the proofs more complicated.

######  Definition 1.2.4.

Let $x=(t,z)\in V$. We say $x$ is _conjugate_ iff $F$ is not a local
diffeomorphism at $x$. The _order_ of conjugacy is the dimension of the kernel
of $dF$. We say $x$ is _a first conjugate vector_ iff no point $(s,z)$, for
$s<t$, is conjugate. We also call the image of the radial vector
$d_{x}F(r_{x})$ a _conjugate vector_ (for $F$) whenever $x$ is conjugate.

In differential geometry, it is more usual to use the term _focal_ instead of
_conjugate_ , when studying the distance function from a hypersurface, but
other authors do otherwise (see for instance [LN]). We have decided to use
this term because our results about the Ambrose conjecture, that will appeal
the most to differential geometers, deal only with the exponential map from a
single point in a Riemannian manifold, while the other results appeal more to
people working in PDEs, which prefer the term _conjugate_.

####  1.2.1. Regular exponential map

The following proposition states some properties of a Finsler exponential map
that correspond approximately to the definition of _regular exponential map_
introduced in [Wa]. The second property is the only one that is not standard,
but we need it to prove the existence of the special coordinates (and only for
that).

######  Proposition 1.2.5.

The exponential map $F$ has the following properties:

  * •

$dF_{x}(r_{x})$ is a non zero vector in $T_{F(x)}M$.

  * •

at every point $x\in V$ there is a basis

$\mathcal{B}=\left\\{v_{1},..,v_{n}\right\\}$

of $T_{x}V$ where $v_{1}=r$ and $v_{n-k+1},..,v_{n}$ span $\ker dF_{x}$, and
such that:

$\mathcal{B}^{\prime}=\left\\{dF(v_{1}),\dots,dF(v_{n-k}),\widetilde{d^{2}F(r\sharp
v_{n-k+1})},\dots\widetilde{d^{2}F(r\sharp v_{n})}\right\\}$

is a basis of $T_{F(x)}M$, where $\widetilde{d^{2}F(r\sharp v_{j})}$ is a
representative of $d^{2}F(r\sharp v_{j})\in T_{F(x)}M/dF(TV_{x})$, for
$n-k+1\leq j\leq n$.

  * •

Any point $x\in V$ has a neighborhood $U$ such that for any ray $\gamma$ (an
integral curve of $r$), the sum of the dimensions of the kernels of $dF$ at
points in $\gamma\cap U$ is constant.

  * •

For any two points $x_{1}\neq x_{2}$ in $V$ with $F(x_{1})=F(x_{2})$,
$dF_{x_{1}}(r_{x_{1}})\neq dF_{x_{2}}(r_{x_{2}})$

###### Proof.

The first three properties follow from the work of Warner [Wa, Theorem 4.5]
for a Finsler exponential map. We emphasize that they are local properties.
The last one follows from the uniqueness property for second order ODEs. We
remark that the second property implies the last one locally. Indeed,
properties 1 and 3 are found in standard textbooks such as [M]. Let us recall
some of the notation in [Wa] and [APS] and show the equivalence of the second
property with his condition (R2) on page 577:

  * •

Second order tangent vectors at a point in an $n$-dimensional manifold are
written in coordinates in the following way ($a_{ij}$ is symmetric):

$\sigma(f)=\sum_{i,j}a_{ij}\frac{\partial^{2}f}{\partial x_{i}\partial
x_{j}}+\sum_{i}b_{i}\frac{\partial f}{\partial x_{i}}$

  * •

$T^{2}_{x}V$ is the set of second order tangent vectors at the point $x$ in
the manifold $V$.

  * •

The second order differential of $F:V\rightarrow M$ at $x$ is the map
$d^{2}_{x}F:T^{2}_{x}V\rightarrow T^{2}_{x}M$ defined by:

$d^{2}_{x}F(\sigma)f=\sigma(f\circ F)$

  * •

The symmetric product $v\sharp w$ of $v\in T_{x}V$ and $w\in T_{x}V$ is a well
defined element of $T^{2}_{x}V/T_{x}V$ with a representative given by the
formula:

$(v\sharp w)f=\frac{1}{2}(v(w(f))+w(v(f)))$

for arbitrary extensions of $v$ to $w$ to vector fields near $x$.

  * •

The map $d^{2}_{x}F$ induces the map $d^{2}F:T^{2}V_{x}/TV_{x}\rightarrow
T^{2}_{F(x)}M/dF(TV_{x})$ by the standard procedure in linear algebra.

  * •

For $x\in V$, $v\in T_{x}V$ and $w\in\ker dF_{x}$, $d^{2}F(v\sharp w)$ makes
sense as a vector in the space $T_{F(x)}M/dF(TV_{x})$. For any extension of
$v$ and $w$, the vector $d^{2}F(v\sharp w)$ is a first order vector.

Thus, our condition is equivalent to property (R2) of Warner:

> At any point $x$ where $\ker dF_{x}\neq 0$, the map
>
> $d^{2}F:T^{2}V_{x}/TV_{x}\rightarrow T^{2}M_{F(x)}/dF(TV_{x})$
>
> sends $\langle r_{x}\rangle\sharp\ker dF_{x}$ isomorphically onto
> $T_{F(x)}M/dF(TV_{x})$.

We recall that $d_{x}F(v)$ can be computed as a Jacobi field when we consider
only the exponential map from a point, but this interpretation is somewhat
diluted when we work with the exponential map from the boundary. Remark.
Warner defines a _regular exponential map_ as any map from $T_{p}M$ into $M$
thast satisfies the properties of proposition 1.2.5. We do not need to work in
that generality, as it does not include any new application.

####  1.2.2. Special coordinates

In order to study the map $F$ more comfortably, we define the _special
coordinates_ , a pair of coordinates near a conjugate point $z$ of order $k$
and its image $F(z)$ that make $F$ specially simple. They can be defined for
any _regular exponential map_. Let $\mathcal{B}=\\{v_{1},\dots,v_{n}\\}$ be
the basis of $T_{z}V$ indicated in the second part of Proposition 1.2.5, and
$\mathcal{B}^{\prime}_{F(z)}$ the corresponding basis at $F(z)\in M$ formed by
vectors $d_{z}F(v_{i}),i\leq n-k$, and $\widetilde{d^{2}_{z}F(v_{1}\sharp
v_{n-k+1})},\dots,\widetilde{d^{2}_{z}F(v_{1}\sharp v_{n})}$. Make a linear
change of coordinates in a neighborhood of $F(z)$ taking
$\mathcal{B}^{\prime}_{F(z)}$ to the canonical basis. The coordinate functions
$F^{i}(x)-F^{i}(z)$ of $F$ for $i\leq n-k$ can be extended to a coordinate
system near $z$ with the help of $k$ functions having $v_{n-k+1},\dots,v_{n}$
as their respective gradients at $z$. In these coordinates $F$ looks:

(1.2.3)
$F(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n-k},F_{z}^{n-k+1}(x),\dots,F_{z}^{n}(x))$

and

  * •

$\frac{\partial}{\partial x_{i}}F_{j}(x^{0})$ is $0$ for any $i$ and $j\geq
n-k+1$,

  * •

$\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial x_{1}}F_{j}(x^{0})$
is $\delta^{i}_{j}$, for $i,j\geq n-k+1$.

  * •

$\frac{\partial}{\partial x_{1}}(x^{0})=r_{x^{0}}$

###  1.3. The Cut Locus

Let $M$ be a Finsler manifold with boundary $\partial M$. We mentioned earlier
that the study of the exponential map from a point or submanifold can be
reduced to a exponential map of a manifold with boundary $\partial M$. The
same principle applies to the cut locus, so we will consider only the cut
locus from the boundary $Cut=Cut_{\partial M}$. It can be defined in several
equivalent ways:

######  Definition 1.3.1.

Let $M$ be a Finsler manifold with boundary $\partial M$:

  * •

For any $p\in\partial M$, let $\gamma_{p}$ be the unit speed geodesic $\gamma$
with initial point in $\partial M$ and initial speed orthogonal to $\partial
M$ (and inner-pointing). $\gamma_{p}$ minimizes the distance between
$\gamma(0)$ and $\gamma(t)$, for small $t$. Define

$t_{cut}(p)=\sup\\{t:d(\gamma_{p}(0),\gamma_{p}(t))=t\\}$

Then

$Cut=\\{\gamma_{p}(t_{cut}(p)):p\in\partial M\\}$

  * •

For a point $p\in M$, let $Q_{p}$ be the set of points $q\in\partial M$ such
that $d(q,p)=d_{\partial M}(p)$. Then $Cut$ is the closure of the set of
points such that $Q_{p}$ has more than one element.

  * •

The function $d_{\partial M}$ is singular exactly when $Q_{p}$ has more than
one element, so we can also define $Cut$ as the closure of the set where the
distance function to $\partial M$ is singular.

  * •

$Cut$ is also the set of points such that either $Q_{p}$ has more than one
element, or $Q_{p}=\\{q\\}$, and $(d(q,p),q)\in V$ is conjugate.

The reader can find the proof of those facts for Riemann manifolds in standard
textbooks in Riemannian geometry (see for example chapter 13 in [dC]). The
proof for Finsler manifolds can be found in [LN], for example. For basic
information about the distance function, such as its differentiabilty, the
reader can use [CS]. Much is known about the set $Cut$:

  * •

$Cut$ is a deformation retract of $M$ (obvious).

  * •

It is the union of a ($n-1$)-dimensional smooth manifold consisting of points
with two minimizing geodesics and a set of Hausdorff dimension at most $n-2$.
This easy but important lemma appears to have been proven at least in [H83],
[IT98], [BL] and [MM], always for the Riemannian case. We give a proof of this
result in lemma 3.2.3 that is also true for Finsler manifolds.

  * •

It is stratified by the dimension of the subdifferential of the _distance to
the boundary_ $\partial d_{\partial M}$. This follows from the properties of
semiconcave functions mentioned in 1.1.2, as $d_{\partial M}$ is semiconcave.
This result can be found in [AAC], and their proof works verbatim for balanced
split locus in our 3.3.3.

  * •

The local homology of the cut locus of $p\in M$ at a point $q$ is related to
the set of minimizing geodesics from $p$ to $q$ (see [Oz] and [H83]).

  * •

The cut locus has finite Hausdorff measure $\mathcal{H}^{n-1}$. This result
can be found in [IT00] for Riemannian manifolds, and in [LN] for Finsler
manifolds. We provide a new proof of this result for Finsler manifolds in 4.4.
M. Castelpietra and L. Rifford also gave a proof of this result that appeared
shortly after the one we present here.

  * •

If all the data is analytic, $Cut$ is a stratified analytic manifold (see
[B77II]). We will not use this result.

  * •

If we add a generic perturbation to $H$ or $M$, Sing becomes a stratified
smooth manifold. Furthermore, for dimension up to $6$, the cut locus is
generically stable, in the sense that adding a small perturbation to the
metric, the new cut locus would still be diffeomorphic to the original one.
This is a very deep result from M. A. Buchner, a student of J. Matter, and it
is beyond the scope of this work to include a proof of his results (see [B]
and [B77]), but we will make use of them in chapter 5.

On the other hand, H. Gluck and D. Singer proved that there are non-
triangulable cut loci in [GS] and, in [GSII], that there are surfaces of
revolution such that the cut locus from any point is non-triangulable. Another
difficulty is mentioned by J. Hebda in [H87]: even though the homotopy of
$Cut$ is known, and even if the cut locus is a simplicial complex, that
simplicial complex may not descend simplicially to one point, and this was a
major obstacle in extending his proof of the Ambrose conjecture for surfaces
to manifolds of higher dimension.

###  1.4. Hamilton-Jacobi equations and Finsler geometry

Here we review the relationship between Hamilton-Jacobi equations and Finsler
geometry. The reader can find more details in [LN], [L] and [CS]. Here $M$ is
a manifold with possibly non-compact boundary. We are interested on solutions
to the system (which we will refer to as a Hamilton-Jacobi Boundary Value
Problem or HJBVP for short):

(1.4.1) $\displaystyle H(p,du(p))\;=$ $\displaystyle 1\quad$ $\displaystyle
p\in M$ (1.4.2) $\displaystyle u(p)\;=$ $\displaystyle g(p)$ $\displaystyle
p\in\partial M$

where $H:T^{\ast}M\rightarrow\mathbb{R}$ is a smooth function that is
$1$-homogeneous and subadditive for linear combinations of covectors lying
over the same point $p$, and $g:\partial M\rightarrow\mathbb{R}$ is a smooth
function that satisfies the compatibility condition:

(1.4.3) $\left|g(p)-g(q)\right|<kd(p,q)\quad\forall p,q\in\partial M$

for some $k<1$. Here $d$ is the distance induced by the Finsler metric
$\varphi$ that is the pointwise dual of the metric in $T^{\ast}M$ given by
$H$:

(1.4.4) $\varphi_{p}(v)=\sup\left\\{\left\langle
v,\alpha\right\rangle_{p}\,:\,\alpha\in
T^{\ast}_{p}M,\,H(p,\alpha)=1\right\\}$

Remark. As mentioned in the introduction, we can ask that $H^{-1}(1)\cap
T^{\ast}_{p}M$ is strictly convex for every $p$ instead of asking that $H$ is
$1$-homogeneous and subadditive for linear combinations of covectors lying
over the same point $p$. The properties are not equivalent for a function $H$,
but the equations that we consider are the same, because the only thing we use
about $H$ is the $1$-level set. If the sets $H^{-1}(1)\cap T^{\ast}_{p}M$ are
convex for every $p$, we can replace $H$ with a new one that is
$1$-homogeneous, subadditive for linear combinations of covectors lying over
the same point $p$, and has the same $1$-level set.

####  1.4.1. Characteristics of the HJBVP

Using the definition 1.1.6 of dual form in Finsler geometry, we can restate
the usual equations for the _characteristic vector field_ at points
$p\in\partial M$:

$\displaystyle\varphi_{p}(X_{p})=1$ $\displaystyle\widehat{X_{p}}|_{T(\partial
M)}=dg$ (1.4.5) $\displaystyle X_{p}\text{ points inwards}$

We define the characteristic vector field as a map $\Gamma:\partial
M\rightarrow TM$, by the formula $\Gamma(p)=X_{p}$. The characteristic curves
are the integral curves of the geodesic vector field in $TM$ with initial
point $\Gamma(z)$ for $z\in\partial M$. The projected characteristics are the
projection to $M$ of the characteristics. A local _classical_ solution $u$ to
the HJBVP can be computed near $\partial M$ following characteristic curves:

######  Definition 1.4.1.

Let $U$ be a neighborhood of $\partial M$ such that every point $q\in U$
belongs to a unique (projected) characteristic contained in $U$ and starting
at a point $p\in\partial M$ (the point $p$ is often called the footpoint of
$q$). The solution by characteristics $u:U\rightarrow\mathbb{R}$ is defined as
follows: if $\gamma:[0,t]\rightarrow M$ is the unique (projected)
characteristic from a point $p\in\partial M$ to $q=\gamma(t)$ that does not
intersect $Sing$, then

$u(q)=g(p)+t$

####  1.4.2. Viscosity solutions of Hamilton-Jacobi equations

The solution found above using characteristic curves is only defined in a
neighborhood of $\partial M$. There is a different notion of solution to these
equations. The inspiration came from the following observation: if we add a
small viscosity term like $-\varepsilon\Delta$ to (1.4.1), that equation
becomes semilinear elliptic, and admits a global solution. So the idea
appeared to add that viscosity term, and later let $\varepsilon$ converge to
$0$. Even though this was the inspiration, it later became clear that it was
better to use a different definition, using comparison functions.

######  Definition 1.4.2.

A function $u:M\rightarrow\mathbb{R}$ is a viscosity subsolution (resp.
supersolution) to the HJBVP given by (1.4.1) and (1.4.2) iff for any $\phi\in
C^{1}(M)$ such that $u-\phi$ has a local maximum at $p$ (resp., a local
minimum), we have:

(1.4.6) $H(p,D\phi(p))\leq 0\qquad\text{(resp. }H(p,D\phi(p))\geq 0\text{)}$

It is a viscosity solution to the HJBVP iff it is both a subsolution and a
supersolution.

The reader can find more details in [L] or [CS]. However, in this thesis we
will not be concerned neither with the inspiration that gave them the name
“viscosity solutions”, nor with the actual definition. We only need to know
that the _unique_ viscosity solution is given by the Lax-Oleinik formula (see
theorem 5.2 in [L]):

(1.4.7) $u(p)=\inf_{q\in\partial M}\left\\{d(p,q)+g(q)\right\\}$

The viscosity solution can be thought of as a way to extend the classical
solution by characteristics to the whole $M$. When $g=0$, the solution (1.4.7)
is the _distance to the boundary_. As we mentioned earlier, the viscosity
solution to a HJBVP is _a semiconcave function_. It is interesting to remark
that a semiconcave function that satifies the equation 1.4.1 at the points at
which it is differentiable is the viscosity solution to the HJBVP.

##  Chapter 2 A new way to look at Cut and Singular Loci

###  2.1. The relation between Finsler geometry and Hamilton-Jacobi BVPs

Let us consider the HJBVP given by (1.4.1) and (1.4.2) when $g=0$. On the one
hand, $\Gamma(p)$ is the inner pointing unit normal to $\partial M$ at $p$. On
the other hand, the viscosity solution $u$ given by (1.4.7) is the distance to
the boundary. This has nice consequences: for example, the singular set of $u$
is a cut locus, and we can apply the various structure results about the cut
locus mentioned in section 1.3. Our intention in this section is to adapt this
result to the case $g>0$. If $\partial M$ is compact, a global constant can be
added to an arbitrary $g$ so that this is satisfied and $S$ is unchanged. We
still require that $g$ satisfies the compatibility condition 1.4.3. Subject to
these conditions, our goal is to show that the Finsler manifold $(M,\varphi)$
can be embedded in a new manifold with boundary $(N,\tilde{\varphi})$ such
that $u$ is the restriction of the unique solution $\tilde{u}$ to the problem

$\displaystyle\tilde{H}(p,d\tilde{u}(p))=1$ $\displaystyle p\in N$
$\displaystyle\tilde{u}(p)=0$ $\displaystyle p\in\partial N$

thus reducing to the original problem ($\tilde{H}$ and $\tilde{\varphi}$ are
dual to one another as in 1.4.4). This allows us to characterize the singular
set of (1.4.7) as a cut locus, which automatically implies that all the
structure results about the cut locus in section 1.3 apply to the more general
case.

######  Definition 2.1.1.

The _indicatrix_ of a Finsler metric $\varphi$ at the point $p$ is the set

$I_{p}=\left\\{v\in T_{p}M\,:\,\varphi(p,v)=1\right\\}$

######  Lemma 2.1.2.

Let $\varphi_{0}$ and $\varphi_{1}$ be two Finsler metrics in an open set $U$,
and let $X$ be a vector field in $U$ such that:

  * •

The integral curves of $X$ are geodesics for $\varphi_{0}$.

  * •

$\varphi_{0}(p,X_{p})=\varphi_{1}(p,X_{p})=1$

  * •

At every $p\in U$, the tangent hyperplanes to the indicatrices of
$\varphi_{0}$ and $\varphi_{1}$ in $T_{p}U$ coincide.

Then the integral curves of $X$ are also geodesics for $\varphi_{1}$

###### Proof.

Let $p$ be a point in $U$. Take bundle coordinates of $T_{p}U$ around $p$ such
that $X$ is one of the vertical coordinate vectors. An integral curve $\alpha$
of $X$ satisfies:

$(\varphi_{0})_{p}(\alpha(t),\alpha^{\prime}(t))=(\varphi_{1})_{p}(\alpha(t),\alpha^{\prime}(t))=1$

because of the second hypothesis. The third hypothesis imply:

$(\varphi_{0})_{v}(\alpha(t),\alpha^{\prime}(t))=(\varphi_{1})_{v}(\alpha(t),\alpha^{\prime}(t))$

So inspection of the geodesic equation:

(2.1.1)
$\varphi_{p}(\alpha(t),\alpha^{\prime}(t))=\frac{d}{dt}\left(\varphi_{v}(\alpha(t),\alpha^{\prime}(t))\right)$

shows that $\alpha$ is a geodesic for $\varphi_{1}$. ∎

######  Corollary 2.1.3.

Let $\varphi$ be a Finsler metric and $X$ a vector field whose integral curves
are geodesics. Then there is a Riemannian metric for which those curves are
also geodesics.

###### Proof.

The Riemannian metric $g_{ij}(p)=\frac{\partial}{\partial
v_{i}v_{j}}\varphi(p,X)$ is related to $\varphi$ as in the preceeding lemma. ∎

######  Lemma 2.1.4.

Let $X$ be a non-zero norm-$1$ geodesic vector field in a Finsler manifold and
$\omega$ its dual differential one-form. Then the integral curves of $X$ are
geodesics if and only if the Lie derivative of $\omega$ in the direction of
$X$ vanishes.

###### Proof.

The integral curves of $X$ are geodesics for $\varphi$ iff they are geodesics
for the Riemannian metric $g_{ij}(p)=\frac{\partial}{\partial
v_{i}v_{j}}\varphi(p,X)$, but the dual one-form to $X$ with respect to both
metrics is the same one-form $\omega$, and the vanishing of
$\mathcal{L}_{X}\omega$ has nothing to do with the metric. We have thus
reduced the problem to a Riemannian metric, when this result is standard:

$\begin{array}[]{lll}\mathcal{L}_{X}(Y)&=&X(w(Y))-\omega([X,Y])\\\
&=&X(\langle X,Y\rangle)-\langle X,D_{X}Y-D_{Y}X\rangle\\\ &=&\langle
D_{X}X,Y\rangle+\langle X,D_{X}Y\rangle-\langle X,D_{X}Y\rangle+\langle
X,D_{Y}X\rangle\\\ &=&\langle D_{X}X,Y\rangle+\frac{1}{2}Y\langle
X,X\rangle\end{array}$

∎

######  Proposition 2.1.5.

Let $M$ be an open manifold with smooth boundary and a Finsler metric
$\varphi$. Let $X$ be a smooth transversal vector field in $\partial M$
pointing inwards (resp. outwards). Then $M$ is contained in a larger open
manifold admitting a smooth extension $\tilde{\varphi}$ of $\varphi$ to this
open set such that the geodesics starting at points $p\in\partial M$ with
initial vectors $X_{p}$ can be continued indefinitely backward (resp. forward)
without intersecting each other.

###### Proof.

We will only complete the proof for a compact manifold with boundary $M$ and
inward pointing vector $X$, as the other cases require only minor
modifications. We start with an arbitrary smooth extension $\varphi^{\prime}$
of $\varphi$ to a larger open set $M_{2}\supset M$. The geodesics with initial
speed $X$ can be continued backwards to $M_{2}$, and there is a small
$\varepsilon$ for which they do not intersect each other for negative values
of time before the parameter reaches $-\varepsilon$. Define

$P:\partial M\times(-\varepsilon,0]\to M_{2},\qquad P(q,t):=\alpha_{q}(t)$

where $\alpha_{q}:(-\varepsilon,0]\to M_{2}$ is the geodesic of
$\varphi^{\prime}$ starting at the point $q\in\partial M$ with initial vector
$X_{q}$. When $p\in U_{\varepsilon}:=\operatorname{Image}(P)$ there is a
unique value of $t$ such that $p=P(q,t)$ for some $q\in\partial M$. We will
denote such $t$ by $d(p)$. Extend also the vector $X$ to $U_{\varepsilon}$ as
$X_{p}=\dot{\alpha_{q}}(t)$ where $p=P(q,t)$. Let
$c:(-\varepsilon,0]\rightarrow[0,1]$ be a smooth function such that

  * •

$c$ is non-decreasing

  * •

$c(t)=1\text{ for }-\varepsilon/3\leq t$

  * •

$c(t)=0\text{ for }t\leq-2\varepsilon/3$

and finally define

$\tilde{X}_{p}=c(d(p))X_{p}$

in the set $U_{\varepsilon}$. Let $\omega_{0}$ be the dual one form of
$\tilde{X}$ with respect to $\varphi$ for points in $\partial M$, and let
$\omega$ be the one form in $U_{\varepsilon}$ whose Lie derivative in the
direction $\tilde{X}$ is zero and which coincides with $\omega_{0}$ in
$\partial M$. Then we take any metric $\varphi^{\prime\prime}$ in
$U_{\varepsilon}$ (which can be chosen Riemannian) such that $\tilde{X}$ has
unit norm and the kernel of $\omega$ is tangent to the indicatrix at
$\tilde{X}$. By lemma 2.1.4, the integral curves of $\tilde{X}$ are geodesics
for $\varphi^{\prime\prime}$. Now let $\rho$ be a smooth function in
$U_{\varepsilon}\cup M$ such that $\rho|_{M}=1$,
$\rho|_{U_{\varepsilon}\setminus U_{\varepsilon/3}}=0$ and $0\leq\rho\leq 1$,
and define the metric:

$\tilde{\varphi}=\rho(p)\varphi(p,v)+(1-\rho(p))\varphi^{\prime\prime}(p,v)$

This metric extends $\varphi$ to the open set $U_{\varepsilon}$ and makes the
integral curves of $\tilde{X}$ geodesics. As the integral curves of $X$ do not
intersect for small $t$, the integral curves of $\tilde{X}$ reach infinite
length before they approach $\partial U_{\varepsilon}$ and the last part of
the statement follows. ∎

Application of this proposition to $M$ and the characteristic, inwards-
pointing vector field $v$ yields a new manifold $N$ containing $M$, and a
metric for $N$ that extends $\varphi$ (so we keep the same letter) such that
the geodesics departing from $\partial M$ which correspond to the
characteristic curves continue indefinitely backwards without intersecting.
This allows the definition, for small $\delta$ of

$\tilde{P}:\partial M\times(-\infty,\delta]\to N,\qquad
P(q,t):=\tilde{\alpha}_{q}(t)$

where $\tilde{\alpha}$ are the geodesics with initial condition $X$, continued
backwards if $t$ is negative. Finally, define
$\tilde{u}:U\rightarrow\mathbb{R}$ by:

(2.1.2) $\tilde{u}(p)=\begin{cases}g(q)+t&p=\tilde{P}(q,t),\quad p\in
N\setminus M\\\ u(p)&p\in M\end{cases}$

We notice that both definitions agree in an _inner_ neighborhood of $\partial
M$, so the function $\tilde{u}$ is a smooth extension of $u$ to $N$.

######  Theorem 2.1.6.

Let $\Lambda=\tilde{u}^{-1}(0)$. Then the following identity holds in
$\left\\{\tilde{u}\geq 0\right\\}$:

(2.1.3) $\tilde{u}(p)=d(\Lambda,p)$

###### Proof.

$\Lambda$ is smooth because it is contained in $N\setminus M$, where
$\tilde{u}$ is smooth and has non-vanishing gradient. In order to show that
$\tilde{u}$ and $d_{\Lambda}$ agree in $U$, we use the uniqueness properties
of viscosity solutions. Let $N$ be the open set where $\tilde{u}>0$. The
distance function to $\Lambda$ is characterized as the unique viscosity
solution to:

  * •

$\tilde{u}=0$ in $\Lambda$

  * •

$H(p,d\tilde{u}(p))=1$ in $N$, in the viscosity sense

Clearly $\tilde{u}$ satisfies the first condition. It also satisfies the
second for points in the set $M$ because it coincides with $u$, and for points
in $N\setminus M$ because $\tilde{u}$ is smooth and $H(p,d\tilde{u}(p))=1$
holds in the classical sense there. ∎

The following fact is well known but we provide a geometric proof.

######  Corollary 2.1.7.

The differential $du$ of the solution by characteristics is Finsler dual to
the tangent to the (projected) characteristics.

###### Proof.

By the above, we can assume that $g=0$. Let $U$ be a neighborhood of $\partial
M$ where a solution by characteristics $u$ is defined, as in 1.4.1. Let $q\in
U$, $X$ be the tangent to the characteristic that goes through $q$, with
footpoint $p$. The claim can be checked easily if $q\in\partial M$, because
both $\tilde{X}$ and $du$ are linear forms and they agree on the hyperplane
$T_{q}\partial M$ and $\Gamma(q)$, by the definition 1.4.1 of the
characteristic vector field. For the rest of points in $U$, we notice that the
level curves of $u$ are Lie parallel with respect to $X$, and so
$L_{X}(du)=0$. But 2.1.4 says that $L_{X}(\tilde{X})=0$, and thus we have two
$1$-forms that agree on $\partial M$ and are parallel with respect to $X$, so
they agree everywhere. ∎

The following theorem is an extension of Theorem 1.1 of [LN]. In this result
$\partial M$ may not be compact.

######  Theorem 2.1.8.

Let $S$ be the closure of the singular set of the viscosity solution to the
following HJBVP:

$\displaystyle H(p,du(p))=1$ $\displaystyle p\in M$ $\displaystyle u(p)=g(p)$
$\displaystyle p\in\partial M$

where $g:\partial M\rightarrow\mathbb{R}$ satisfies the usual compatibility
condition 1.4.3. If $\mu$ is the function whose value at $p\in\partial M$ is
the distance to $S$ along the unique characteristic departing from $q$, then

  1. (1)

$\mu$ is Lipschitz.

  2. (2)

If in addition $\partial M$ is compact, then the $(n-1)$-dimensional Hausdorff
measure of $S\cap K$ is finite for any compact $K$.

  3. (3)

In general, $S$ is a Finsler cut locus from the boundary of some Finsler
manifold, so all the regularity results for cut loci apply to $S$ (see section
1.3).

###### Proof.

The first part follows immediately from Theorem 2.1.6 and Theorem 1.1 in [LN].
The second is an easy consequence of the first, while the last is contained in
the results of this section. ∎

Remark. The regularity hypothesis on $M$ can be softened. In order to apply
the results in [LN], it is enough that $M$ is $C^{2,1}$, which implies that
$\Lambda$ is $C^{2,1}$.

###  2.2. Split locus and balanced split locus

We study a Hamilton-Jacobi equation given by (1.4.1) and (1.4.2) in a
$C^{\infty}$ compact manifold with boundary $M$, with the hypothesis stated
there. Let $Sing$ be the closure of the singular set of the viscosity solution
$u$ to aHamilton-Jacobi BVP. $Sing$ has a key property: any point in
$M\setminus Sing$ can be joined to $\partial M$ by a unique characteristic
curve that does not intersect $Sing$. A set with this property is said to
_split $M$ along characteristics of the HJBVP_ or simply to _split_ $M$ for
short. Once characteristic curves are known, if we replace $Sing$ by any set
$S$ that splits $M$, we can use the value of $u$ that the characteristics
carry along with them, to obtain a function, defined in $M\setminus S$ with
some resemblance to the viscosity solution (see definition 2.2.5). Looking at
the cut locus from this new perspective, we wonder what distinguishes the cut
locus from all the other sets that split $M$.

######  Definition 2.2.1.

For a set $S\subset M$, let $A(S)\subset V$ be the set of all $x=(t,z)\in V$
such that $F(s,z)\notin S,\,\forall\,0\leq s<t$. We say that a set $S\subset
M$ _splits_ $M$ iff $F$ restricts to a bijection between $A(S)$ and
$M\setminus S$.

Whenever $S$ splits $M$, we can define a vector field $R_{p}$ in $M\setminus
S$ to be $dF_{x}(r_{x})$ for the unique $x$ in $A(S)$ such that $F(x)=p$.

######  Definition 2.2.2.

For a point $a\in S$, we define the _limit set_ $R_{a}$ as the set of vectors
in $T_{a}M$ that are limits of sequences of the vectors $R_{p}$ defined above
at points $p\in M\setminus S$.

######  Remark 2.2.3.

If $S$ is a cut locus, the set $R_{p}$ is the set of all vectors tangent to
the minimizing geodesics from $p$ to $\partial M$.

######  Definition 2.2.4.

If $S$ splits $M$, we also define a set $Q_{p}\subset V$ for $p\in M$ by

$Q_{p}=\left(F|_{\overline{A(S)}}\right)^{-1}(p)$

The following relation holds between the sets $R_{p}$ and $Q_{p}$:

$R_{p}=\left\\{dF_{x}(r_{x}):x\in Q_{p}\right\\}$

######  Definition 2.2.5.

If $S$ splits $M$, we can define a real-valued function $h$ in $M\setminus S$
by setting:

$h(p)=g(z)+t$

where $(t,z)$ is the unique point in $A(S)$ with $F(t,z)=p$.

If we start with the viscosity solution $u$ to the Hamilton-Jacobi equations,
and let $S=Sing$ be the closure of the set where $u$ is not $C^{1}$, then $S$
splits $M$. If we follow the above definition involving $A(S)$ to get a new
function $h$, then we find $h=u$.

######  Definition 2.2.6.

A set $S$ that splits $M$ is a split locus iff

$S=\overline{\left\\{p\in S:\quad\sharp R_{p}\geq 2\right\\}}$

The role of this condition is to restrict $S$ to its _essential_ part. A set
that merely splits $M$ could be too big: actually $M$ itself splits $M$. The
following lemma may clarify this condition.

######  Lemma 2.2.7.

A set $S$ that splits $M$ is a split locus if and only if $S$ is closed and it
has no proper closed subsets that split $M$.

###### Proof.

The “if” part is trivial, so we will only prove the other implication. Assume
$S$ is a split locus and let $S^{\prime}\subset S$ be a closed set splitting
$M$. Let $q\in S\setminus S^{\prime}$ be a point with $\sharp R_{q}\geq 2$.
Since $S^{\prime}$ is closed, there is a neighborhood of $q$ away from
$S^{\prime}$; so, if $\gamma_{1}$ is a segment of a geodesic in $M\setminus
S^{\prime}$ joining $\partial M$ with $q$, there is a point $q_{1}$ in
$\gamma_{1}$ lying beyond $q$. Furthermore, we can choose the point $q_{1}$
not lying in $S$, so there is a second geodesic $\gamma_{2}$ contained in
$M\setminus S\subset M\setminus S^{\prime}$ from $\partial M$ to $q_{1}$. As
$q\in S$, we see $\gamma_{2}$ is necessarily different from $\gamma_{1}$,
which is a contradiction if $S^{\prime}$ splits $M$. Therefore we learn
$S^{\prime}\supset\left\\{p\in S:\quad\sharp R_{p}\geq 2\right\\}$, so
$S=\overline{\left\\{p\in S:\quad\sharp R_{p}\geq 2\right\\}}\subset
S^{\prime}$. ∎

Finally, we introduce the following more restrictive condition (see 1.1.1 for
the definition of $v_{p}(q)$, the _vector from $p$ to $q$_, and 1.1.6 for the
Finsler dual of a vector).

######  Definition 2.2.8.

We say a split locus $S\subset M$ is _balanced_ for given $M$, $H$ and $g$ (or
simply that it is balanced if there is no risk of confusion) iff for all $p\in
S$, all sequences $p_{i}\to p$ with $v_{p_{i}}(p)\to v\in T_{p}M$, and any
sequence of vectors $X_{i}\in R_{p_{i}}\to X_{\infty}\in R_{p}$, then

$w_{\infty}(v)=\max\left\\{w(v),\text{ $w$ is dual to some $R\in
R_{p}$}\right\\}$

where $w_{\infty}$ is the dual of $X_{\infty}$.

Figure 2.2.1. An arbitrary split locus and a balanced split locus

###  2.3. Balanced property of the Finsler cut locus

In this section we show that the cut locus of a Finsler exponential map is a
balanced set. We provide two proofs, none of which is original. Although the
hypothesis look different, our result 2.1.6 show that they are equivalent. The
first proof is the same as in lemma 2.1 in [IT00], but we adapt it so that it
also works for Finsler manifolds, where angles are not defined.

######  Proposition 2.3.1.

The cut locus of a Finsler manifold $M$ with boundary is a balanced split
locus. Moreover, for $p$, $p_{n}$, $v$ and $X_{\infty}$ as in the definition
of a balanced split locus, we have

$\lim_{n\rightarrow\infty}\dfrac{d(\partial M,p)-d(\partial
M,p_{n})}{d(p,p_{n})}=w_{\infty}(v)$

###### Proof.

The cut locus $S$ splits $M$, as follows from the first definition of cut
locus in 1.3.1. It is also a split locus, as follows from the second
definition of cut locus. Next we show that $S$ is balanced. Take any $Y\in
R_{p}$, and let $\gamma$ be the minimizing geodesic segment joining $\partial
M$ to $p$ with speed $Y$ at $p$. Take any point $q\in\gamma$ that lies in a
convex neighborhood of $p$ and use the triangle inequality to get:

$d(\partial M,p)-d(\partial M,p_{n})\geq d(q,p)-d(q,p_{n})$

The first variation formula yields, for a constant $C$:

$d(q,p)-d(q,p_{n})\geq w(v_{p_{n}}(p))d(p_{n},p)-Cd(p,p_{n})^{2}$

and we get:

$\liminf_{n\rightarrow\infty}\dfrac{d(\partial M,p)-d(\partial
M,p_{n})}{d(p,p_{n})}\geq w(X)$

for any $w$ that is dual to a vector in $R_{p}$. Then consider $X_{\infty}$,
let $\gamma$ be the minimizing geodesic segment joining $\partial M$ to $p$
with speed $X_{\infty}$ at $p$, and let $\gamma_{n}$ be the minimizing
geodesic segment joining $\partial M$ to $p_{n}$ with speed $X_{n}$ at
$p_{n}$. Take points $q_{n}$ in $\gamma_{n}$ that lie in a fix convex
neighborhood of $p$. Again:

$d(\partial M,p)-d(\partial M,p_{n})\leq d(q_{n},p)-d(q_{n},p_{n})$

while the first variation formula yields, for a constant $C$:

$d(q_{n},p)-d(q_{n},p_{n})\leq w(v_{p_{n}}(p))d(p_{n},p)+Cd(p,p_{n})^{2}$

and thus:

$\limsup_{n\rightarrow\infty}\dfrac{d(\partial M,p)-d(\partial
M,p_{n})}{d(p,p_{n})}\leq w_{\infty}(X)$

This proves the claim that $S$ is balanced. ∎

We give now another proof that relates the balanced condition to the notion of
semiconcave functions, which is now common in the study of Hamilton-Jacobi
equations. More precisely, we simply translate theorem 3.3.15 in the book [CS]
to our language to get the following lemma:

######  Lemma 2.3.2.

The closure of the singular set of the viscosity solution to (1.4.1) and
(1.4.2) is a balanced split locus.

###### Proof.

Let $u$ be the viscosity solution to (1.4.1) and (1.4.2), and let $Sing$ be
the closure of its singular set. We leave to the reader the proof that $Sing$
is a split locus (otherwise, recall it is a cut locus). It is well known that
$u$ is semiconcave (see for example [CS, 5.3.7]). The superdifferential
$D^{+}u(p)$ of $u$ at $p$ is the convex hull of the set of limits of
differentials of $u$ at points where $u$ is $C^{1}$ (see [CS, 3.3.6]). At a
point where $u$ is $C^{1}$, the dual of the speed vector of a characteristic
is the differential of $u$. Thus, the superdifferential at $p$ is the convex
hull of the duals to the vectors in $R_{p}$. We deduce:

$\max\left\\{w(v),\text{ $w$ is dual to some $R\in
R_{p}$}\right\\}=\max\left\\{w(v),w\in D^{+}u(p)\right\\}$

Given $p\in M$, and $v\in T_{p}M$, the _exposed face_ of $D^{+}u(p)$ in the
direction $v$ is given by:

$D^{+}(p,v)=\\{\tilde{w}\in D^{+}u(p):\tilde{w}(v)\leq w(v)\;\forall w\in
D^{+}u(p)\\}$

The balanced condition can be rephrased in these terms as:

> Let $p_{i}\rightarrow p\in S$ be a sequence with $v_{p_{i}}(p)\to v\in
> T_{p}M$, and let $w_{i}\in D^{+}u(p_{i})$ be a sequence converging to $w\in
> D^{+}u(p)$. Then $w\in D^{+}u(p,-v)$

which is exactly the statement of theorem [CS, 3.3.15], with two minor
remarks:

  1. (1)

The condition is restricted to points $p\in S$. At points in $M\setminus S$,
the balanced condition is trivial.

  2. (2)

In the balanced condition, we use the vectors $v_{p_{i}}(p)$ from $p_{i}$ to
$p$, contrary to the reference [CS]. Thus the minus sign in the statement.

∎

In the light of this new proof, we can regard the balanced condition as a
differential version of the semiconcavity condition. A semiconcave function
that is a solution to (1.4.1) is also a viscosity solution (see [CS, 5.3.1]).
We will later study if the solution of (1.4.1) built by characteristics using
a balanced split loci is also a viscosity solution.

##  Chapter 3 Local structure of cut and singular loci up to codimension 3

Our main result in this chapter is a local description of the cut locus around
any point of the cut locus _except for a set of Hausdorff dimension $n-3$_
(see Theorem 3.1.2). Actually, our structure results hold for the more general
_balanced split loci_ (recall that in 2.3.1 and 2.1.8 we showed that cut loci,
and singular sets of solutions to HJ equations, are balanced split loci).
Working in this generality complicates some proofs and, in particular, we have
to prove some results for _balanced split loci_ that are long known to be true
for cut loci. However, we need to actually prove the structure results for
balanced split loci for the applications to chapter 4, and all the new proofs
of old facts are either short or interesting for their own sake.

###  3.1. Statements of results

For the results of this chapter, $M$ is a $C^{\infty}$ Finsler manifold with
_compact_ boundary $\partial M$, but $M$ itself need not be compact. $S\subset
M$ is a balanced split locus. Recall 2.2.2 for the definition of $R_{p}$.

####  3.1.1. Results

Our main result asserts that we can avoid conjugate points of order $2$ and
above if we neglect a set of Hausdorff dimension $n-3$:

######  Theorem 3.1.1 (Conjugate points of order $2$).

There is a set $N\subset S$ of Hausdorff dimension at most $n-3$ such that for
any $p\in S\setminus N$ and $x\in V$ such that $F(x)=p$ and $d_{x}F(r_{x})\in
R_{p}$:

$\dim(\ker d_{x}F)\leq 1$

Combining this new result with previous ones in the literature, we are able to
provide the following description of a cut locus. All the extra results
required for the proof of this result will be proved in this chapter.

######  Theorem 3.1.2 (The cut locus up to $\mathcal{H}$-codimension 3).

Let $S$ be either the cut locus of a point or submanifold in a Finsler
manifold or the closure of the singular locus of a solution of 1.4.1 and
1.4.2. Then $S$ consists of the following types of points :

  * •

Cleave points: Points at which $R_{p}$ consists of two non-conjugate vectors.
The set of cleave points is a smooth hypersurface;

  * •

Edge points: Points at which $R_{p}$ consists of exactly one conjugate vector
of order 1. This is a set of Hausdorff dimension at most $n-2$;

  * •

Degenerate cleave points: Points at which $R_{p}$ consists of two vectors,
such that one of them is conjugate of order 1, and the other may be non-
conjugate or conjugate of order 1. This is a set of Hausdorff dimension at
most $n-2$;

  * •

Crossing points: Points at which $R_{p}$ consists of non-conjugate and
conjugate vectors of order 1, and $R^{\ast}_{p}$ spans an affine subspace of
dimension $2$. This is a rectifiable set of dimension at most $n-2$;

  * •

Remainder: A set of Hausdorff dimension at most $n-3$;

In the next chapter, we will provide more detailed descriptions of how does a
balanced split loci looks near each of these different points (see 4.6.2,
4.6.3, 4.6.5, 4.6.6 and 4.6.7). In the next chapter and also in section 6.5 we
show applications of this result, but we believe it can also be useful in
other contexts. For instance, stochastic processes on manifolds is often
studied on the complement of the cut locus from a point, and then the results
have to be adapted to take care of the situation when the process hits the cut
locus (see [BL]). Brownian motion, for example, almost never hits a set with
null $\mathcal{H}^{n-2}$ measure, but will almost surely hit any set with
positive $\mathcal{H}^{n-2}$ measure, so we think our result can be useful in
that field.

####  3.1.2. Examples

We provide examples of Riemannian manifolds and exponential maps which
illustrate our results. First, consider a solid ellipsoid with two equal
semiaxis and a third larger one. This is a 3D manifold with boundary, and the
geodesics starting at the two points that lie further away from the center
have a first conjugate vector of order $2$ while remaining minimizing up to
that point. This example shows that our bound on the Hausdorff dimension of
the points in the cut locus with a minimizing geodesic of order $2$ cannot be
improved. Second, consider the surface of an ellipsoid with three different
semiaxis (or any generic surface as in [B], with metric close to the standard
sphere) and an arbitrary point on it. It is known that in the tangent space
the set of first conjugate points is a closed curve $C$ bounding the origin,
and at most of these points the kernel of the exponential map is transversal
to the curve $C$. More explicitely, the set $C^{\ast}$ of points of $C$ where
it is not transversal is finite. Consider then the product $M$ of two such
ellipsoids. The exponential map onto $M$ has a conjugate point of order $2$ at
any point in $(C\setminus C^{\ast})\times(C\setminus C^{\ast})$, and the
kernel of the exponential map is transversal to the tangent to $C\times C$.
Thus the image of the set of conjugate points of order $2$ is a smooth
manifold of codimension $2$. This example shows that theorem 3.1.1 does not
hold for the image of all the conjugate points of order $2$, and only holds
for the _minimizing_ conjugate points. Finally, recall the construction in
[GS], where the authors build a riemannian surface whose cut locus is not
triangulable. Their example shows that the set of points with a conjugate
minimizing geodesic can have infinite $\mathcal{H}^{n-2}$ measure. A similar
construction replacing the circle in their construction with a 3d ball shows
that the set of points with a minimizing geodesic conjugate of order $2$ can
have infinite $\mathcal{H}^{n-3}$ measure.

####  3.1.3. Relation to previous results in the literature

Our structure theorem generalizes a standard result that has been proven
several times by mathematicians from different fields (see for example
[BL]111Although there is a mistake in their proof, [H87], [MM] and [IT98]):

> A cut locus in a Riemannian manifold is the union of a smooth
> $(n-1)$-dimensional manifold $\mathcal{C}$ and a set of zero
> $(n-1)$-dimensional Hausdorff measure (actually, a set of Hausdorff
> dimension at most $n-2$). The set $\mathcal{C}$ consists of cleave points,
> which are joined to the origin or initial submanifold by exactly two
> minimizing geodesics, both of which are non-conjugate.

This result follows from 3.1.2, since the union of edge, degenerate cleave,
and crossing points is a set of Hausdorff dimension at most $n-2$. The
statement quoted above follows from lemmas 3.2.3, 3.3.2 and 3.3.3 only.
Theorem 3.1.1 is not necessary if a description is needed only up to
codimension $2$. The proof of the three lemmas is simple and has many features
in common with earlier results on the cut locus. In a previous paper, A. C.
Mennucci studied the singular set of solutions to the HJ equations with only
$C^{k}$ regularity. Under this hypothesis, the set $S\setminus\mathcal{C}$ may
have Hausdorff dimension strictly between $n-1$ and $n-2$ (see [Me]). We work
only in a $C^{\infty}$ setting, and under this stronger condition, the set
$S\setminus\mathcal{C}$ has always Haussdorf dimension at most $n-2$. Our
result 3.1.2 uses the theory of singularities of semi-concave functions that
can be found for example in [AAC]. Though their result can be applied to a
Finsler manifold, we had to give a new proof that applies to balanced split
loci instead of just the cut locus.

###  3.2. Conjugate points in a balanced split locus

In this section we prove Theorem 3.1.1. Throughout this section, $M$, $r$, $V$
and $F$ are as in section 1.2 and $S$ is a balanced split locus as defined in
2.2.8.

######  Definition 3.2.1.

A singular point $x\in V$ of the map $F$ is an _A2_ point if $ker(dF_{x})$ has
dimension $1$ and is transversal to the tangent to the set of conjugate
vectors.

Remark. Warner shows in [Wa] that the set of conjugate points of order $1$ is
a smooth (open) hypersurface inside $V$, and that for adequate coordinate
functions in $V$ and $M$, the exponential has the following normal form around
any A2 point,

(3.2.1)
$\displaystyle(x_{1},x_{2},\dots,x_{m})\longrightarrow(x_{1}^{2},x_{2},\dots,x_{m})$

######  Proposition 3.2.2.

For any $p\in M$ and $X\in R_{p}$, the vector $X$ is not of the form
$dF_{x}(r)$ for any A2 point $x$.

###### Proof.

The proof is by contradiction. Let $p\in S$ be such that $R_{p}$ contains an
A2 vector $Z=dF_{c}(r_{c})$, for $c\in Q_{p}$. By the normal form (3.2.1), we
see there is a neighborhood $U$ of $c$ such that no other point in $U$ maps to
$p$. Furthermore, in a neighborhood $B$ of $p$ the image of the conjugate
vectors is a hypersurface $H$ such that all points at one side (call it
$B_{1}$) have two preimages of $F|_{U}$, all points at the other side $B_{2}$
of $H$ have no preimages, and points at $H$ have one preimage, whose
corresponding vector is A2-conjugate. It follows that $Z$ is isolated in
$R_{p}$. We notice there is a sequence of points $p_{n}\rightarrow p$ in
$B_{2}$ with vectors $Y_{n}\in R_{p_{n}}$ such that $Y_{n}\rightarrow Y\neq
X$. Thus $R_{a}$ does not reduce to $Z$. The vector $Z$ is tangent to $H$, so
we can find a sequence of points $p_{n}\in B_{2}$ approaching $p$ such that

$\lim_{n\rightarrow\infty}v_{p_{n}}(p)=Z$

We can find a subsequence $p_{n_{k}}$ of the $p_{n}$ and vectors $X_{k}\in
R_{p_{n_{k}}}$ such that $X_{k}$ converges to some $X_{\infty}\in R_{p}$. By
the above, $X_{\infty}$ is different from $Z$, but $\hat{Z}(X)<1=\hat{Z}(Z)$
(where $\hat{Z}$ is the dual form to $Z$), so the balanced property is
violated. ∎

The following is the analogous to Theorem 3.1.1 for conjugate points of order
$1$.

######  Proposition 3.2.3 (Conjugate points of order $1$).

There is a set $N\subset S$ of Hausdorff dimension $n-2$ such that for all
$p\in S\setminus N$, $R_{p}$ does not contain conjugate vectors.

###### Proof.

The proof is identical to the proof of lemma 2 in [IT98] for a cut locus, but
we include it here for completeness. First of all, at the set of conjugate
vectors of order $k\geq 2$ we can apply directly the Morse-Sard-Federer
theorem (see [F]) to show that the image of the set of conjugate cut vectors
of order $k\geq 2$ has Hausdorff dimension at most $n-2$. Let $Q$ be the set
of conjugate vectors of order $1$ (recall it is a smooth hypersurface in $V$).
Let $G$ be the set of conjugate vectors such that the kernel of $dF$ is
tangent to the conjugate locus. Apply the Morse-Sard-Federer theorem again to
the map $F|_{Q}$ to show that the image of $G$ has Hausdorff dimension at most
$n-2$. Finally, the previous result takes cares of the $A2$ points. ∎

We now turn to the main result of this paper: we state and prove Theorem 3.2.4
which has 3.1.1 as a direct consequence.

######  Theorem 3.2.4.

Let $M$, $V$, $F$ and $r$ be as in section 1.2, and let $S$ be a balanced
split locus (2.2.8). The set of conjugate points of order $2$ in $V$
decomposes as the union of two subsets $Q_{2}^{1}$ and $Q_{2}^{2}$ such that:

  * •

No point in $Q_{2}^{1}$ maps under $dF$ to a vector in any of the $R_{a}$ (in
set notation: $dF(Q_{2}^{1})\cap\left(\cup R_{p}\right)=\emptyset$)

  * •

The image under $F$ of $Q_{2}^{2}$ has Hausdorff dimension at most $n-3$.

###### Proof.

Let $z$ be a conjugate point of order $2$ and take special coordinates at
$U_{z}$ near $z$. In the special coordinates near $z$ (see 1.2.2), $F$ is
written:

(3.2.2)
$F(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n-2},F_{z}^{n-1}(x),F_{z}^{n}(x))$

for some functions $F_{z}^{n-1}$ and $F_{z}^{n}$, and $x=(x_{1},\dots,x_{n})$
in a neighborhood $U_{z}$ of $z$ with $F(0,\dots,0)=(0,\dots,0)$. The Jacobian
of $F$ is:

$JF=\left[\begin{array}[]{ccccc}1&\ldots&0&\ast&\ast\\\
\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&\ldots&1&\ast&\ast\\\
0&\ldots&0&\frac{\partial F_{z}^{n-1}}{\partial x_{n-1}}&\frac{\partial
F_{z}^{n}}{\partial x_{n-1}}\\\ 0&\ldots&0&\frac{\partial
F_{z}^{n-1}}{\partial x_{n}}&\frac{\partial F_{z}^{n}}{\partial
x_{n}}\end{array}\right]$

A point $x$ is of order $2$ if and only if the $2\times 2$ submatrix for the
$x_{n-1}$ and $x_{n}$ coordinates (and the corresponding coordinates in
$F(U_{z})$: $y_{n-1}$ and $y_{n}$) vanish:

(3.2.3) $\left[\begin{array}[]{cc}\frac{\partial F_{z}^{n-1}}{\partial
x_{n-1}}&\frac{\partial F_{z}^{n}}{\partial x_{n-1}}\\\ \frac{\partial
F_{z}^{n-1}}{\partial x_{n}}&\frac{\partial F_{z}^{n}}{\partial
x_{n}}\end{array}\right]=0$

Near a point of order $2$, we write:

$F_{z}^{n-1}(x)=x_{1}x_{n-1}+q(x_{n-1},x_{n})+T^{n-1}(x)$
$F_{z}^{n}(x)=x_{1}x_{n}+r(x_{n-1},x_{n})+T^{n}(x)$

where $q(x_{n-1},x_{n})$ and $r(x_{n-1},x_{n})$ are the quadratic terms in
$x_{n-1}$ and $x_{n}$ in a Taylor expansion, and $T$ consists of terms of
order $\geq 3$ in $x_{n-1}$ and $x_{n}$, and terms of order $\geq 2$ with at
least one $x_{i},i\leq n-2$. The nature of the polynomials $q$ and $r$ in the
special coordinates at $z$ will determine whether $z$ is in $Q_{2}^{1}$ or in
$Q_{2}^{2}$. We have the following possibilities:

  1. (1)

either $q$ or $r$ is a sum of squares of homogeneous linear functions in
$x_{n-1}$ and $x_{n}$ (possibly with a global minus sign).

  2. (2)

both $q$ and $r$ are products of distinct linear functionals (equivalently,
they are difference of squares). Later on, we will split this class further
into three types: 2a, 2b and 2c.

  3. (3)

one of $q$ and $r$ is zero, the other is not.

  4. (4)

both $q$ and $r$ are zero.

We set $Q_{2}^{1}$ to be the points of type 1 and 2c, and $Q_{2}^{2}$ to be
the points of type 2a, 3 and 4. Points of type 2b do not appear under the
hypothesis of this theorem.

###### Type 1.

The proof is similar to Proposition 3.2.2. Assume $z=(0,\dots,0)$ is of type
1. If, say, $q$ is a sum of squares, then in the set
$\\{x_{1}=a,x_{2}=\dots=x_{n-2}=0\\}\cap F(U_{z})$, $x_{n-1}$ will reach a
minimum value that will be greater than $-Ca^{2}$ for some $C>0$. We learn
there is a sequence $p^{k}=(t^{k},0,\dots,x_{n-2},-(C+1)(t^{k})^{2},0)$, for
$t^{k}\nearrow 0$, approaching $(0,\dots,0)$ with incoming speed
$(1,0,\dots,0)$ and staying in the interior of the complement of $F(U_{z})$
for $k$ large enough. Pick up any vectors $V_{k}\in R_{p^{k}}$ converging to
some $V_{0}$ (passing to a subsequence if necessary). Then $V_{0}$ is
different from $(1,0,\dots,0)\in R_{0}$, and

$\widehat{V_{0}}\left((1,\dots,0)\right)<\widehat{(1,\dots,0)}\left((1,\dots,0)\right)=1$

violating the balanced condition.

###### Type 2 and 3.

If a point is of type 2 or 3, we can assume $q\neq 0$. Before we proceed, we
change coordinates to simplify the expression of $F$ further. Consider a
linear change of coordinates near $x$ that mixes only the $x_{n-1}$ and
$x_{n}$ coordinates.

$\left(\begin{array}[]{c}x^{\prime}_{n-1}\\\
x^{\prime}_{n}\end{array}\right)=A\cdot\left(\begin{array}[]{c}x_{n-1}\\\
x_{n}\end{array}\right)$

followed by the linear change of coordinates near $p$ that mixes only the
$y_{n-1}$ and $y_{n}$ coordinates with the inverse of the matrix above:

$\left(\begin{array}[]{c}y^{\prime}_{n-1}\\\
y^{\prime}_{n}\end{array}\right)=A^{-1}\cdot\left(\begin{array}[]{c}y_{n-1}\\\
y_{n}\end{array}\right)$

Straightforward but tedious calculations show that there is a matrix $A$ such
that the map $F$ has the following expression in the coordinates above:

$F(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n-2},x_{1}x_{n-1}+(x_{n-1}^{2}-x^{2}_{n}),x_{1}x_{n}+r(x_{n-1},x_{n}))+T$

In other words, we can assume $q(x_{n-1},x_{n})=(x_{n-1}^{2}-x^{2}_{n})$. Fix
small values for all $x_{i}$ for $2\leq i\leq n-2$. At the origin, $JF$ is a
diagonal matrix with zeros in the positions $(n-1,n-1)$ and $(n,n)$. We recall
that $z$ is conjugate of order $2$ iff the submatrix (3.2.3) vanishes. This
submatrix is the sum of

(3.2.4) $\displaystyle\begin{bmatrix}x_{1}+2x_{n-1}&r_{x_{n-1}}\\\
-2x_{n}&x_{1}+r_{x_{n}}\end{bmatrix}$

and some terms that either have as a factor one of the $x_{i}$ for $2\leq
i\leq n-2$, or are quadratic in $x_{n-1}$ and $x_{n}$. We want to show that,
near points of type 3 and some points of type 2, all conjugate points of order
$2$ are contained in a submanifold of codimension $3$. The claim will follow
if we show that the gradients of the four entries span a $3$-dimensional space
at points in $U$. For convenience, write $r(x_{n-1},x_{n})=\alpha
x_{n-1}^{2}+\beta x_{n-1}x_{n}+\gamma x_{n}^{2}$. It is sufficient that the
matrix with the partial derivatives with respect to $x_{i}$ for
$i\in\\{1,n-1,n\\}$ of the four entries have rank $3$:

$A=\begin{bmatrix}1&2&0\\\ 0&0&-2\\\ 0&2\alpha&\beta\\\
1&\beta&2\gamma\end{bmatrix}$

The claim holds if all $x_{i}$ are small, for $i\not\in\\{1,n-1,n\\}$, unless
$\alpha=0$ and $\beta=2$. This covers points of type $3$. We say a point of
type 2 has type 2a if the rank of the above matrix is $3$. Otherwise, the
polynomial $r$ looks:

$r(x_{n-1},x_{n})=2x_{n-1}x_{n}+\gamma
x_{n}^{2}=2x_{n}(x_{n-1}+\frac{\gamma}{2}x_{n})$

We say a point of type 2 has type 2b if $r$ has the above form and
$-1<\frac{\gamma}{2}<1$. We will show that there are integral curves of $r$
arbitrarily close to the one through $z$ without conjugate points near $z$,
which contradicts property 3 of exponential maps in Proposition 1.2.5. Take a
ray $t\rightarrow\zeta_{x_{n}}(t)$ passing through a point
$(0,\dots,0,x_{n})$. The determinant of 3.2.3 along the ray is:

$\begin{array}[]{rl}d(t)&=\frac{\partial F_{z}^{n-1}}{\partial
x_{n-1}}(\zeta(t))\frac{\partial F_{z}^{n}}{\partial
x_{n}}(\zeta(t))-\frac{\partial F_{z}^{n}}{\partial
x_{n-1}}(\zeta(t))\frac{\partial F_{z}^{n-1}}{\partial x_{n}}(\zeta(t))\\\
&=t^{2}+t(4x_{n-1}+2\gamma x_{n})+(4x_{n-1}^{2}+4\gamma
x_{n-1}x_{n}+4x_{n}^{2})+R_{3}(x_{n},t)\\\ &=(t+2x_{n-1}+\gamma
x_{n})^{2}+(4-\gamma^{2})x_{n}^{2}+R_{3}(x_{n},t)\\\ &\geq
c(t^{2}+x_{n}^{2})+R_{3}(x_{n},t)\end{array}$

for a remainder $R_{3}$ of order $3$. Thus there is a $\delta>0$ such that for
any $x_{n}\neq 0$ and $|t|<\delta$, $|x_{n}|<\delta$, $\zeta_{x_{n}}(t)$ is
not a conjugate point. We have already dealt with points of type 3, 2a and 2b.
Now we turn to the rest of points of type 2 (type 2c). We have either
$\frac{\gamma}{2}\geq 1$ or $\frac{\gamma}{2}\leq-1$. We notice that
$x_{n-1}^{2}-x_{n}^{2}\leq 0$ iff $|x_{n-1}|\leq|x_{n}|$, but whenever
$|x_{n-1}|\leq|x_{n}|$, the sign of $r(x_{n-1},x_{n})$ is the sign of
$\gamma$. Thus the second order part of $F$ maps $U$ into the complement of
points with negative second coordinate and whose third coordinate has the
opposite sign of $\gamma$. A similar argument as the one for type 1 points
yields a contradiction with the balanced condition. If, for example,
$\gamma\geq 2$, none of the following points

$x^{k}=(t^{k},-(C+1)(t^{k})^{2},-(C+1)(t^{k})^{2},..0,)$

is in $F(U)$, for $t^{k}\rightarrow 0$. But then we can carry a vector other
than $(1,0,\dots,0)$ as we approach $F(x_{0})$.

###### Type 4.

Let $z$ be a conjugate point of order $2$. We show that the image of the
points of type 4 inside $U_{z}$ has Hausdorff dimension at most $n-3$. $U_{z}$
is an open set around an arbitrary point $z$ of order $2$, and thus the result
follows. First, we find that for any point $x$ of type 4, we have
$d^{2}_{x}F(v\sharp w)=0$ for all $v,w\in\ker d_{x}F$, making the computation
in the special coordinates at $x\in U_{z}$ (see section 1.2.1 for the
definition of $d^{2}F$). Then we switch to the special coordinates around $z$.
In these coordinates, the kernel of $dF$ at $x$ is generated by
$\frac{\partial}{\partial x_{n-1}}$ and $\frac{\partial}{\partial x_{n}}$.
Thus $\frac{\partial^{2}F_{z}^{n-1}}{\partial x_{i}x_{j}}=0$ for $i,j\geq n-1$
at any point $x\in U_{z}$ of type 4. The set of conjugate points of order $2$
is contained in the set $H\\!=\\!\\{\frac{\partial F_{z}^{n-1}}{\partial
x_{n-1}}(x)\\!=\\!0\\}$. This set is a smooth hypersurface: the second
property in 1.2.5 implies that $\frac{\partial^{2}F_{z}^{n-1}}{\partial
x_{1}x_{n-1}}\neq 0$ at points of $H$. At every conjugate point of type 4, the
kernel of $dF$ is contained in the tangent to $H$. Thus conjugate points of
type 4 are conjugate points of the restriction of $F$ to $H$. The Morse-Sard-
Federer theorem applies, and the image of the set of points of type 4 has
Hausdorff dimension $n-3$.∎

###### Proof of Theorem 3.1.1.

Follows immediately from the above, setting $N=F(Q^{2}_{2})$. ∎

###  3.3. Structure up to codimension 3

This section contains the proof of 3.1.2, splitted into several lemmas. All of
them are known for cut loci in riemannian manifolds, but we repeat the proof
so that it applies to balanced split loci in Finsler manifolds.

######  Definition 3.3.1.

We say $p\in S$ is a _cleave_ point iff $R_{p}$ has two elements $X^{1}$ and
$X^{2}$, with $(p,X^{1})=(F(y_{1}),dF_{y_{1}}(r_{y_{1}}))$ and
$(p,X^{2})=(F(y_{2}),dF_{y_{2}}(r_{y_{2}}))$, and both $dF_{y_{1}}$ and
$dF_{y_{2}}$ are non-singular. In other words, $p\in S$ is a cleave point iff
$R_{p}$ consists of two non-conjugate vectors.

######  Proposition 3.3.2.

$\mathcal{C}$ is a $(n-1)$-dimensional manifold.

###### Proof.

Let $p=F(y_{1})=F(y_{2})$ be a cleave point, with
$R_{p}\\!=\\!\\{dF_{y_{1}}(r),dF_{y_{2}}(r)\\}$. We can find a small
neighborhood $U$ of $p$ so that the following conditions are satisfied:

  1. (1)

$U$ is the diffeomorphic image of neighborhoods $U_{1}$ and $U_{2}$ of the
points $y_{1}$ and $y_{2}$. Thus, the two smooth vector fields
$X^{1}_{q}=dF|_{U_{1}}(r)$ and $X^{2}_{q}=dF|_{U_{2}}(r)$ are defined in
points $q\in U$.

  2. (2)

At all points $q\in U$, $R_{q}\subset\\{X^{1}_{q},X^{2}_{q}\\}$. Other vectors
must be images of the vector $r$ at points not in $U_{1}$ or $U_{2}$, and if
they accumulate near $p$ we could find a subsequence converging to a vector
that is neither $X_{1}$ nor $X_{2}$. We reduce $U$ if necessary to achieve the
property.

  3. (3)

Let $H$ be an hypersurface in $U_{1}$ passing through $y_{1}$ and transversal
to $X_{1}$, and let $\tilde{H}=F(H)$. We define local coordinates $p=(z,t)$ in
$U$, where $z\in\tilde{H}$ and $t\in\mathbb{R}$ are the unique values for
which $p$ is obtained by following the integral curve of $X^{1}$ that starts
at $x$ for time $t$. $U$ is a cube in these coordinates.

We will show that $S$ is a graph in the coordinates $(z,t)$. Let $A_{i}$ be
the set of points $q$ for which $R_{q}$ contains $X^{i}_{q}$, for $i=1,2$. By
the hypothesis, $S=A_{1}\cap A_{2}$. Every tangent vector $v$ to $S$ at $q\in
S$ (in the sense of 1.1.2), satisfies the following property (where $\hat{X}$
is the dual covector to a vector $X\in TM$.):

$\hat{X}^{i}(v)=\max_{Y\in R_{p}}\hat{Y}(v)$

which in this case amounts to $\hat{X}^{1}(v)=\hat{X}^{2}(v)$, or

$v\in\ker(\hat{X}^{1}-\hat{X}^{2})$

We can define in $U$ the smooth distribution
$D=\ker(\hat{X}^{1}-\hat{X}^{2})$. $S$ is a closed set whose approximate
tangent space is contained in $D$. We first claim that for all $z$, there is
at most one time $t_{0}$ such that $(z,t_{0})$ is in $S$. If $(z,t)$ is in
$A_{1}$, $R_{(z,t)}$ contains $X^{1}$ and, unless $(z,s)$ is contained in
$A_{1}$ for $s$ in an interval $(t-\varepsilon,t)$, we can find a sequence
$(z_{n},t_{n})$ converging to $(z,t)$ with $t_{n}\nearrow t$ and carrying
vectors $X^{2}$. The incoming vector is $X^{1}$, but

$\tilde{X^{2}}(X^{1})<\tilde{X^{1}}(X^{1})=1$

which contradicts the balanced property. Analogously, if $R_{(z,t)}$ contains
$X^{2}$ there is an interval $(t,t+\varepsilon)$ such that $(z,s)$ is
contained in $A_{2}$ for all $s$ in the interval. Otherwise there is a
sequence $(z_{n},t_{n})$ converging to $(z,t)$ with $t_{n}\searrow t$ and
carrying vectors $X^{1}$. The incoming vector is $-X^{1}$, but

$-1=\tilde{X^{1}}(-X^{1})<\tilde{X^{2}}(-X^{1})$

which is again a contradiction. The claim follows easily. We show next that
the set of $p$ for which there is a $t$ with $(z,t)\in S$ is open and closed
in $\Gamma$, and thus $S$ is the graph of a function $h$ over $\Gamma$. Take
$(z,t)\in U\cap S$ and choose a cone $D_{\varepsilon}$ around $D_{p}$. We can
assume the cone intersects $\partial U$ only in the $z$ boundary. There must
be a point in $S$ of the form $(z^{\prime},t^{\prime})$ inside the cone for
all $z^{\prime}$ sufficiently close to $z$: otherwise there is either a
sequence $(z_{n},t_{n})$ approaching $(z,t)$ with $t_{n}>h_{+}(z)$ ($h$ being
the upper graph of the cone $D_{\varepsilon}$) and carrying vectors $X^{1}$ or
a similar sequence with $t_{n}<h_{-}(z)$ and carrying vectors $X^{2}$. Both
options violate the balanced condition. Closedness follows trivially from the
definition of $S$. Define $t=h(z)$ whenever $(z,t)\in S$. The tangent to the
graph of $h$ is given by $D$ at every point, thus $S$ is smooth and indeed an
integral maximal submanifold of $D$. ∎

Remark. It follows from the proof above that there cannot be any balanced
split locus unless $D$ is integrable. This is not strange, as the sister
notion of cut locus does not make sense if $D$ is not integrable. We recall
that the orthogonal distribution to a geodesic vector field is parallel for
that vector field, so the distribution is integrable at one point of the
geodesic if and only if it is integrable at any other point. In particular, if
the vector field leaves a hypersurface orthogonally (which is the case for a
cut locus) the distribution $D$ (which is the difference of the orthogonal
distributions to two geodesic vector fields) is integrable. It also follows
from 2.1.8 that the characteristic vector field in a Hamilton-Jacobi problem
has an integrable orthogonal distribution. Remark. In the next chapter we
study whether a balanced split locus is actually a cut locus. The proof of the
above lemma showed there is a unique sheet of cleave points near a given point
in a balanced split loci.

######  Proposition 3.3.3.

The set of points $p\in S$ where $co\,(R^{\ast}_{p})$ has dimension $k$ is
$(n-k)$-rectifiable.

###### Proof.

Throughout the proof, let $\hat{X}$ be the dual covector to the vector $X\in
TM$. Let $p_{n}$ be a sequence of points such that $co\,(R^{\ast}_{p_{n}})$
contains a $k$-dimensional ball of radius greater than $\delta$. Suppose they
converge to a point $p$ and $v_{p_{n}}(p)$ converges to a vector $\eta$. We
take a neighborhood $U$ of $p$ and fix product coordinates in $\pi^{-1}(U)$ of
the form $U\times\mathbb{R}^{n}$. Then, we extract a subsequence of $p_{n}$
and vectors $X_{n}^{1}\in R_{p_{n}}$ such that $X_{n}^{1}$ converge to a
vector $X^{1}$ in $R_{p}$. Outside a ball of radius $c\delta$ at
$\hat{X}_{n}^{1}$, where $c$ is a fixed constant and $n>>0$, there must be
vectors in $R_{p_{n}}$, and we can extract a subsequence of $p_{n}$ and
vectors $X_{n}^{2}$ converging to a vector $X^{2}$ such that $\hat{X}^{2}$ is
at a distance at least $c\delta$ of $\hat{X}^{1}$. Iteration of this process
yields a converging sequence $p_{n}$ and $k$ vectors

$X_{n}^{1},..,X_{n}^{k}\in R_{p_{n}}$

converging to vectors

$X^{1},..,X^{k}\in R_{p}$

such that the distance between $\hat{X}^{k}$ and the linear span of
$\hat{X}^{1},..\hat{X}^{k-1}$ is at least $c\delta$, so that $coV^{\ast}_{p}$
contains a $k$-dimensional ball of radius at least $c^{\prime}\delta$. The
balanced property implies that the $\hat{X}^{j}$ evaluate to the same value at
$\eta$, which is also the maximum value of the $\hat{Z}(\eta)$ for a vector
$Z$ in $R_{p}$. In other words, the convex hull of the $\hat{X}^{j}$ belong to
the face of $R^{\ast}_{p}$ that is exposed by $\eta$. If $co\,R_{p}^{\ast}$ is
$k$-dimensional, $\eta$ belongs to

$\begin{array}[]{rl}\left(co\,R_{p}^{\ast}\right)^{\perp}=&\left\\{v\in
T_{p}M\>:\quad\langle w,v\rangle\text{ is constant for }w\in
co\,R_{p}^{\ast}\right\\}\\\\[8.61108pt] =&\left\\{v\in
T_{p}M\>:\quad\langle\hat{X},v\rangle\text{ is constant for }X\in
R_{p}\right\\}\end{array}$

which is a $n-k$ dimensional subspace. Let $\Sigma^{k}_{\delta}$ be the set of
points $p\in S$ for which $co\,R_{p}^{\ast}$ is $k$-dimensional and contains a
$k$-dimensional ball of radius greater than or equal to $\delta$. We have
shown that all tangent directions to $\Sigma^{k}_{\delta}$ at a point $p$ are
contained in a $n-k$ dimensional subspace. We can apply theorem 3.1 in [AAC]
to deduce $\Sigma^{k}_{\delta}$ is $n-k$ rectifiable, so their union for all
$\delta>0$ is rectifiable too. ∎

##  Chapter 4 Balanced split sets and Hamilton-Jacobi equations

###  4.1. Introduction

In this chapter we consider the Hamilton-Jacobi boundary value problem (1.4.1)
and (1.4.2) in a compact set $M$. A local _classical_ solution can be computed
near $\partial M$ following _characteristic_ curves as in section 1.4.1. A
unique _viscosity solution_ is given by the Lax-Oleinik formula (1.4.7). The
viscosity solution can be thought of as a way to extend the classical solution
to the whole $M$. Recall from section 2.3 that the singular set $Sing$ is a
_balanced split locus_. This notion was inspired originally by the paper
[IT00], but is also related to the notion of _semiconcave_ functions that is
now common in the study of Hamilton-Jacobi equations (see section 2.3). Our
goal in this chapter is to determine whether there is a unique balanced split
locus. In the cases when this is not true, we also give an interpretation of
the multiple balanced split loci.

####  4.1.1. Outline

In section 4.2 we state our results, give examples, and comment on possible
extensions. Section 4.3 gathers some of the results from the literature we
will need, and includes a few new lemmas that we use later. Section 4.4
contains our proof that the _distance to a balanced split locus_ and _distance
to the $k$-th conjugate point_ are Lipschitz. Section 4.5 contains the proof
of the main theorems, modulo a result that is proved in section 4.6. This last
section also features detailed descriptions of a balanced split set at each of
the points in the classification in theorem 3.1.2.

###  4.2. Statement of results.

####  4.2.1. Results

For _fixed_ $M$, $H$ and $g$ satisfying the conditions stated earlier, there
is always at least one balanced split locus, namely the singular set of the
solution of (1.4.1) and (1.4.2). In general, there might be more than one
balanced split loci, depending on the topology of $M$. Our first theorem
covers a situation where there is uniqueness.

######  Theorem 4.2.1.

Assume $M$ is simply connected and $\partial M$ is connected. Then there is a
unique balanced split locus, which is the singular locus of the solution of
(1.4.1) and (1.4.2).

The next theorem removes the assumption that $\partial M$ is connected, and
uniqueness goes away:

######  Theorem 4.2.2.

Assume $M$ is simply connected and $\partial M$ has several connected
components. Let $S\subset M$ be a balanced split locus. Then $S$ is the
singular locus of the solution of (1.4.1) and (1.4.2) with boundary data $g+a$
where the function $a$ is constant at each connected component of $\partial
M$.

The above theorem describes precisely all the balanced split loci in a
situation where there is non-uniqueness. If $M$ is not simply connected, the
balanced split loci are more complicated to describe. We provide a somewhat
involved procedure using the universal cover of the manifold. However, the
final answer is very natural in the light of the examples.

######  Theorem 4.2.3.

There exists a bijection between balanced split loci for given $M$, $H$ and
$g$ and an open subset of the homology space $H^{1}(M,\partial M)$ containing
zero.

In fact, this theorem follows immediately from the next, where we construct
such bijection:

######  Theorem 4.2.4.

Let $\widetilde{M}$ be the universal cover of $M$, and lift both $H$ and $g$
to $\widetilde{M}$. Let $a:[\partial\widetilde{M}]\rightarrow\mathbb{R}$ be an
assignment of a constant to each connected component of
$\partial\widetilde{M}$ that is equivariant for the action of the automorphism
group of the covering and such that $\widetilde{g}(z)+a(z)$ satisfies the
compatibility condition (1.4.3) in $\widetilde{M}$. Then the singular locus
$\widetilde{S}$ of the solution $\widetilde{u}$ to:

$\widetilde{H}(x,d\widetilde{u}(x))=1\quad x\in\widetilde{M}$
$\widetilde{u}(x)=\widetilde{g}(x)+a(z)\quad x\in\partial\widetilde{M}$

is invariant by the automorphism group of the covering, and its quotient is a
set $S$ that is a balanced split locus for $M$, $H$ and $g$. Furthermore:

  1. (1)

The procedure above yields a bijection between balanced split loci for given
$M$, $H$ and $g$ and _equivariant compatible_ functions
$a:[\partial\widetilde{M}]\rightarrow\mathbb{R}$.

  2. (2)

Among the set of equivariant functions
$a:[\partial\widetilde{M}]\rightarrow\mathbb{R}$ (that can be identified
naturally with $H^{1}(M,\partial M)$), those compatible correspond to an open
subset of $H^{1}(M,\partial M)$ that contains $0$.

###### Remark.

The space $H^{1}(M,\partial M)$ is dual to $H_{n-1}(M)$ by Lefschetz theorem.
The proof of the above theorems rely on the construction from $S$ of a
$(n-1)$-dimensional current $T_{S}$ that is shown to be closed and thus
represents a cohomology class in $H_{n-1}(M)$. The proof of the above theorem
also shows that the map sending $S$ to the homology class of $T_{S}$ is a
bijection from the set of balanced split loci onto a subset of $H_{n-1}(M)$.
In order to prove these theorems we will make heavy use of some _structure
results_ for balanced split loci. To begin with, we start with the results
from the previous chapter, specifically theorem 3.1.2. In the last section, we
prove _new structure results_ in order to improve the description of balanced
split loci near each of these types of points (see 4.6.2, 4.6.3, 4.6.5, 4.6.6
and 4.6.7). We also study some very important functions for the study of the
cut locus. Recall the global coordinates in $V$ given by $z\in\partial M$ and
$t\in\mathbb{R}$. Let $\lambda_{j}(z)$ be the value of $t$ at which the
geodesic $s\rightarrow\Phi(s,z)$ has its $j$-th conjugate point (counting
multiplicities), or $\infty$ if there is no such point. Let $\rho_{S}:\partial
M\rightarrow\mathbb{R}$ be the minimum $t$ such that $F(t,z)\in S$.

######  Lemma 4.2.5.

All functions $\lambda_{j}:\partial M\rightarrow\mathbb{R}$ are Lipschitz
continuous.

######  Lemma 4.2.6.

The function $\rho_{S}:\partial M\rightarrow\mathbb{R}$ is Lipschitz
continuous if $S$ is balanced.

Both results were proven in [IT00] for Riemannian manifolds, and the second
one was given in [LN]. Thus, our results are not new for a cut locus, but the
proof is different from the previous ones and may be of interest. We have
recently known of another proof that $\rho$ and $\lambda_{1}$ are Lipschitz
([CR]).

####  4.2.2. Examples

Take as $M$ any ring in a euclidean $n$-space bounded by two concentric
spheres. Solve the Hamilton-Jacobi equations with $H(x,p)=|p|$ and $g=0$. The
solution is the distance to the spheres, and the cut locus is the sphere
concentric to the other two and equidistant from each of them. However, any
sphere concentric to the other two and lying between them is a balanced split
set, so there is a one parameter family of split balanced sets. When $n>2$,
this situation is a typical application of 4.2.2. In the $n=2$ case, there is
also only one free parameter, which is in accord with 4.2.4, as the rank of
the $H_{1}$ homology space of the ring is one. For a more interesting example,
we study balanced split sets with respect to a point in a euclidean torus. We
take as a model the unit square in the euclidean plane, centered at the
origin, with its borders identified. It is equivalent to study the distance
with respect to a point in this euclidean torus, or the solution to Hamilton-
Jacobi equations with respect to a small disc centered at the origin with the
Hamiltonian $H(p)=|p|$ and $g=0$.

Figure 4.2.1. Balanced split set in a torus

A branch of cleave points (see 3.1.2) must keep constant the difference of the
distances from either sides (recall the proof of prop 3.3.2, or read the
beginning of section 4.5). Moving to the covering plane of the torus, we see
they must be segments of hyperbolas. A balanced split locus is the union of
the cleave segments and a few triple or quadruple points. The set of all
balanced split loci is a $2$-parameter family, as predicted by our theorem
4.2.4.

###  4.3. Preliminaries

####  4.3.1. Lagrangian submanifolds of $T^{\ast}M$

######  Definition 4.3.1.

The canonical symplectic form in $T^{\ast}M$ is given in canonical coordinates
by

$\sum_{i}dp_{i}\wedge dq_{i}$

A submanifold $L\subset T^{\ast}M$ is Lagrangian iff the restriction of the
canonical symplectic form to $L$ vanishes.

Let $D$ be the duality homeomorphism between $TM$ and $T^{\ast}M$ induced by
the Finsler metric as in definition 1.1.6 ($D$ is actually a $C^{\infty}$
diffeomorphism away from the zero section). We define a map:

(4.3.1) $\Delta(t,z)=D(\Phi_{t}(\Gamma(z)))$

and a subset of $T^{\ast}M$:

(4.3.2) $\Theta=\Delta(V)=D(W)$

where $\Phi_{t}$ is the geodesic flow in $TM$. This is a smooth
$n$-submanifold of $T^{\ast}M$ with boundary. It is a standard fact that, for
a smooth function $u:M\rightarrow\mathbb{R}$, the graph of its differential
$du$ is a Lagrangian submanifold of $T^{\ast}M$, for the canonical symplectic
structure in $T^{\ast}M$. The subset of $\Theta$ corresponding to small $t$ is
the graph of the differential of the solution by characteristics $u$ to the HJ
equations. Indeed, all of $\Theta$ is a lagrangian submanifold of $T^{\ast}M$
when $\Gamma$ comes from an exponential map. As we have seen, this covers
HJBVPs as well. We can also carry over the geodesic vector field from $TM$
into $T^{\ast}M$ (outside the zero sections). This vector field in $T^{\ast}M$
is tangent to $\Theta$. Then, as we follow an integral curve $\gamma(t)$
within $\Theta$, the tangent space to $\Theta$ describes a curve $\lambda(t)$
in the bundle $G$ of lagrangian subspaces of $T^{\ast}M$. It is a standard
fact that the vector subspace $\lambda(t)\subset T^{\ast}_{\gamma(t)}M$
intersects the vertical subspace of $T^{\ast}_{\gamma(t)}M$ in a non-trivial
subspace for a discrete set of times. We will review this fact, in elementary
terms, and prove a lemma that will be important for the proof of lemma 4.2.6.
Let $\eta(t)$ be an integral curve of $r$ with $x_{0}=\eta(0)$ a conjugate
point of order $k$. In special coordinates near $x_{0}$, for $t$ close to $0$,
the differential of $F$ along $\eta$ has the form:

$dF(\eta(t))=\begin{pmatrix}I_{n-k}&0\\\
\ast&\ast\end{pmatrix}=\begin{pmatrix}I_{n-k}&0\\\
0&0\end{pmatrix}+t\begin{pmatrix}0&0\\\
0&I_{k}\end{pmatrix}+\begin{pmatrix}0&0\\\ \ast&E(t)\end{pmatrix}$

where $|E|<\varepsilon$, with $E=0$ if $\gamma(0)=x_{0}$. Let $w\in\ker
dF(\eta(t_{1}))$ and $v\in\ker dF(\eta(t_{2}))$ be unit vectors in the kernel
of $dF$ for $t_{1}<t_{2}$ close to $0$. It follows that both $v$ and $w$ are
spanned by the last $k$ coordinates. We then find:

$0=w\cdot dF(\eta(t_{2}))\cdot v-v\cdot dF(\eta(t_{1}))\cdot
w=(t_{2}-t_{1})w\cdot v+w(E(t_{2})-E(t_{1}))v$

and it follows (for some $t_{1}<t^{\ast}<t_{2}$):

$(t_{2}-t_{1})w\cdot v<2\varepsilon|w||v|(t_{2}-t_{1})$

and so:

(4.3.3) $w\cdot v<2\varepsilon$

This also shows that the set of $t$’s such that $dF(\eta(t))$ is singular is
discrete. Say the point $x_{0}=(z_{0},t_{0})$ is the $j$-th conjugate point
along the integral curve of $r$ through $x_{0}$ from $z_{0}$, and recall that
it is of order $k$ as conjugate point. As $z$ moves towards $z_{0}$, all
functions $\lambda_{j}(z),\dots,\lambda_{j+k}(z)$ converge to $t_{0}$. Let
$z_{i}$ be a sequence of points converging to $z_{0}$ such that the integral
curve through $z_{i}$ meets its $k$ conjugate points near $z_{0}$ at $M$
linear subspaces (e.g. $\lambda_{j}(z_{i})=\dots=\lambda_{j+k_{1}}(z_{i})$;
$\lambda_{j+k_{1}+1}(z_{i})=\dots=\lambda_{j+k_{2}}(z_{i})$; …;
$\lambda_{j+k_{M-1}+1}(z_{i})=\dots=\lambda_{j+k_{M}}(z_{i})$). we get the
following result (see also lemma 1.1 in [IT00]):

######  Lemma 4.3.2.

The subspaces $\ker d_{(\lambda_{j+k_{l}}(z_{i}),z_{i})}F$ for $l=1,\dots,M$
converge to orthogonal subspaces of $\ker d_{(\lambda_{j}(z_{0}),z_{0})}F$,
for the standard inner product in the special coordinates at the point
$(\lambda_{j}(z_{0}),z_{0})$.

####  4.3.2. A useful lemma

######  Lemma 4.3.3.

Let $U$ be an open set in $\mathbb{R}^{n}$, $A\subset U$ a proper open set,
$C^{+}\subset\mathbb{R}^{n}$ an open cone, $V\subset U$ an arbitrary open set
and $\varepsilon>0$ such that at any point $q\in\partial A\cap V$, we have
$(q+C^{+})\cap(q+B_{\varepsilon})\subset A$. Then $\partial A\cap V$ is a
Lipschitz hypersurface. Moreover, for any vector $X\in C^{+}$, take
coordinates so that $X=\frac{\partial}{\partial x_{1}}$. Then $\partial A\cap
V$ is a graph $S=\\{(h(x_{2},..,x_{n}),x_{2},$ $..,x_{n})\\}$ for a Lipschitz
function $h$.

###### Proof.

Choose the vector $X\in C^{+}$ and coordinate system in the statement. Assume
$X$ has norm $1$, so that $q+tX\in q+B_{t}$ for small positive $t$. Take any
point $p\in\partial A\cap V$. We claim that all points
$p+t\frac{\partial}{\partial x_{1}}$ for $0<t<\varepsilon$ belong to $A$, and
all points $p+t\frac{\partial}{\partial x_{1}}$ for $-\varepsilon<t<0$ belongs
to $U\setminus A$. Indeed, there cannot be a point
$p+t\frac{\partial}{\partial x_{1}}\in A$ for $-\varepsilon<t<0$ because the
set $(p+t\frac{\partial}{\partial x_{1}})+(C^{+}\cap B_{\varepsilon})$ would
contain an open neighborhood of $p$, which contains points not in $A$. In
particular, there is at most one point of $\partial A\cap V$ in each line with
direction vector $\frac{\partial}{\partial x_{1}}$. Take two points
$q_{1},q_{2}\in\mathbb{R}^{n-1}$ sufficiently close and consider the lines
$L_{1}=\\{(t,q_{1}),t\in\mathbb{R}\\}$ and
$L_{2}=\\{(t,q_{2}),t\in\mathbb{R}\\}$. Assume there is a $t_{1}$ such that
$(t_{1},q_{1})$ belongs to $\partial A$. If there is no point of $\partial A$
in $L_{2}$ then either all points of $L_{2}$ belong to $A$ or they belong to
$U\setminus A$. Both of these options lead to a contradiction if
$((t_{1},q_{1})+C^{+})\cap((t_{1},q_{1})+B_{\varepsilon})\cap
L_{2}\neq\emptyset$ (this condition is equivalent to
$K|q_{1}-q_{2}|<\varepsilon$ for a constant $K$ that depends on $C^{+}$ and
the choice of $X\in C^{+}$ and the coordinate system). Thus there is a point
$(t_{2},q_{2})\in\partial A$. For the constant $K$ above and $t\geq
t_{1}+K|q_{1}-q_{2}|$, the point $(t,q_{2})$ lies in the set
$(t_{1},q_{1})+C^{+}$, so we have

$t_{2}<t_{1}+K|q_{1}-q_{2}|$

The points $q_{1}$ and $q_{2}$ are arbitrary, and the lemma follows. ∎

Remark. We are working in a paper about some limitations of the technique of
3d printing known as fused desposition modeling. The above lemma is used to
prove that all current 3d printers using this technique will print pieces that
are specially fragile in some directions.

####  4.3.3. Some generalities on HJ equations.

######  Lemma 4.3.4.

For fixed $M$ and $H$, two functions $g,g^{\prime}:\partial
M\rightarrow\mathbb{R}$ have the same characteristic vector field in $\partial
M$ iff $g^{\prime}$ can be obtained from $g$ by addition of a constant at each
connected component of $\partial M$.

###### Proof.

It follows from (1.4.1) that $g$ and $g^{\prime}$ have the same characteristic
vector field at all points if and only $dg=dg^{\prime}$ at all points. ∎

For our next definition, observe that given $M$, $H$ and $g$, we can define a
map $\tilde{u}:V\rightarrow\mathbb{R}$ by $\tilde{u}(t,z)=t+g(z)$.

######  Definition 4.3.5.

We say that a function $u:M\rightarrow\mathbb{R}$ is _made from
characteristics_ iff $u|_{\partial M}=g$ and $u$ can be written as
$u(p)=\tilde{u}\circ s$ for a (not necessarily continuous) section $s$ of
$F:V\rightarrow M$.

###### Remark.

In the paper [Me], the same idea is expressed in different terms: all
characteristics are used to build a multi-valued solution, and then some
criterion is used to select a one-valued solution. The criterion used there is
to select the characteristic with the minimum value of $\tilde{u}$.

######  Lemma 4.3.6.

The viscosity solution to (1.4.1) and (1.4.2) is the unique continuous
function that is made from characteristics.

###### Proof.

Let $h$ be a function made from characteristics, and $u$ be the viscosity
solution given by formula (1.4.7). Let $Sing$ be the closure of the singular
set of $u$. Take a point $z\in\partial M$. Define:

$t^{\ast}_{z}=\sup\left\\{t\geq 0:\;h(F(\tau,z))=u(F(\tau,z))\;\forall
0\leq\tau<t\right\\}$

Let $p=F(t^{\ast}_{z},z)$. Assume for simplicity that $h(p)=u(p)$. Claim:
$t^{\ast}_{z}<\rho_{Sing}(z)$ implies $h$ is discontinuous at
$F(t^{\ast}_{z},z)$. Proof of the claim: Assume that
$t^{\ast}_{z}<\rho_{Sing}(z)$ and $h$ is continuous at $F(t^{\ast}_{z},z)$ for
some $z\in\partial M$. As $t^{\ast}_{z}<\rho_{Sing}(z)\leq\lambda_{1}(z)$,
there is an open neighborhood $O$ of $(t^{\ast}_{z},z)$ such that $F|_{O}$ is
a diffeomorphism onto a neighborhood of $p=F(t^{\ast}_{z},z)$. By hypothesis,
there is a sequence $t_{n}\rightarrow t^{\ast}_{z}$ and $p_{n}=F(t_{n},z)$
such that $h(p_{n})\neq u(p_{n})$. As $h$ is built from characteristics using
a section $s$, we have
$h(p_{n})=\tilde{u}(s(p_{n}))=\tilde{u}((s_{n},y_{n}))=s_{n}+g(y_{n})$, for
$(s_{n},y_{n})\neq(t_{n},z)$. For $n$ big enough, the point $(s_{n},y_{n})$
does not belong to $O$, as $(t_{n},z)$ is the only preimage of $p_{n}$ in $O$.
As $h(p_{n})\rightarrow h(p)$, and $\partial M$ is compact, we deduce the
$s_{n}$ are bounded. We can take a subsequence of $(s_{n},y_{n})$ converging
to $(s_{\infty},y_{\infty})\not\in O$. So we have
$p=F(t^{\ast}_{z},z)=F(s_{\infty},y_{\infty})$. If $p\not\in Sing$, we deduce
that
$\lim_{n\rightarrow\infty}h(p_{n})=\tilde{u}(s_{\infty},y_{\infty})>h(p)=u(p)=\tilde{u}(t^{\ast}_{z},z)$,
so $h$ is discontinuous at $p$. Using the claim, we conclude the proof: if $h$
is continuous, then $\rho_{Sing}(z)\leq t^{\ast}_{z}$ for all $z\in\partial
M$, and $u=h$, as any point in $M$ can be expressed as $F(t,z)$ for some $z$,
and some $t\leq\rho_{Sing}(z)$. ∎

We will need later the following version of the same principle:

######  Lemma 4.3.7.

Let $S$ be a split locus, and $h$ be the function associated to $S$ as in
definition 2.2.5. If $\rho_{S}$ is continuous, and $h$ can be extended to $M$
so that it is continuous except for a set of null $\mathcal{H}^{n-1}$ measure,
then $S=Sing$.

###### Proof.

Define

$Y_{0}=\left\\{z\in\partial M:\;h(F(t,z))\neq u(F(t,z))\;\text{ for some
}t\in[0,\rho_{Sing}(z))\right\\}$

By the claim in the previous lemma, $Y_{0}$ is contained in:

$Y=\left\\{z\in\partial M:\;h\text{ discontinuous at }F(t,z)\;\text{ for some
}t\in[0,\rho_{Sing}(z))\right\\}$

Let $A=A(Sing)$ be the set in definition 2.2.1. The map $F$ restricts to a
diffeomorphism from $A$ onto $M\setminus Sing$. The set $Y$ can be expressed
as:

$Y=\pi_{2}\circ(F|_{A})^{-1}\left(\\{p\in M\setminus Sing:\;h\text{
discontinuous at }p\\})\right)$

and thus by the hypothesis has null $\mathcal{H}^{n-1}$ measure. Therefore,
$\partial M\setminus Y_{0}$ is dense in $\partial M$. We claim now that
$S\subset Sing$. To see this, let $p\in S\setminus Sing$. Then
$p=F(t^{\ast},z^{\ast})$ for a unique $(t^{\ast},z^{\ast})\in A$. It follows
$\rho_{S}(z^{\ast})\leq t^{\ast}<\rho_{Sing}(z^{\ast})$. As $\rho_{S}$ is
continuous, $\rho_{S}(z)<\rho_{Sing}(z)$ holds for all $z$ in a neighborhood
of $z^{\ast}$ in $\partial M$ and, in particular, for some $z\in\partial
M\setminus Y_{0}$. This is a contradiction because, for
$\rho_{S}(z)<t<\rho_{Sing}(z)$, $h(F(t,z))=\tilde{u}(t^{\prime},z^{\prime})$
for $(t^{\prime},z^{\prime})\neq(t,z)$, and $t<\rho_{Sing}(z)$ implies
$h(F(t,z))=\tilde{u}(t^{\prime},z^{\prime})>\tilde{u}(t,z)=u(F(t,z))$, forcing
$z\in Y_{0}$. We deduce $S=Sing$ using lemma 2.2.7 and the fact that $Sing$ is
a split locus. ∎

###  4.4. $\rho_{S}$ is Lipschitz

In this section we study the functions $\rho_{S}$ and $\lambda_{j}$ defined
earlier. The fact that $\rho_{S}$ is Lipschitz will be of great importance
later. The definitions and the general approach in this section follow [IT00],
but our proofs are shorter, provide no precise quantitative bounds, use no
constructions from Riemannian or Finsler geometry, and work for Finsler
manifolds, thus providing a new and shorter proof for the main result in [LN].
The proof that $\lambda_{j}$ are Lipschitz functions was new for Finsler
manifolds when we published the first version of the preprint of this paper.
Since then, the paper [CR] has appeared which shows that $\lambda_{1}$ is
actually semi-concave.

###### Proof of 4.2.5.

It is immediate to see that the functions $\lambda_{j}$ are continuous, since
this is property (R3) of Warner (see [Wa, pp. 577-578 and Theorem 4.5]). Near
a conjugate point $x^{0}$ of order $k$, we can take special coordinates as in
section 1.2.2:

$F(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n-k},F_{n-k+1},\dots,F_{n})$

Conjugate points near $x$ are the solutions of

$d(x_{1},\dots,x_{n})=det(dF)=\sum_{\sigma}(-1)^{\sigma}\frac{\partial
F_{\sigma(n-k+1)}}{\partial x_{n-k+1}}\dots\frac{\partial
F_{\sigma(n)}}{\partial x_{n}}=0$

From the properties of the special coordinates, we deduce that:

(4.4.1) $D^{\alpha}d(0)=0\qquad\forall|\alpha|<k$

and

$\frac{\partial^{k}}{\partial x_{1}^{k}}d=1$

We can use the preparation theorem of Malgrange (see [GG]) to find real valued
functions $q$ and $l_{i}$ in an open neighborhood $U$ of $x$ such that
$d(x)\neq 0$ and:

$q(x_{1},\dots,x_{n})d(x_{1},\dots,x_{n})=x_{1}^{k}+x_{1}^{k-1}l_{1}(x_{2},\dots,x_{n})+\dots+l_{k}(x_{2},\dots,x_{n})$

and we deduce from (4.4.1) that

(4.4.2) $D^{\alpha}l_{i}(0)=0\qquad\forall|\alpha|<i$

which implies

(4.4.3)
$|l_{i}(x_{2},\dots,x_{n})|<\bar{C}\max\\{|x_{2}|,\dots,|x_{n}|\\}^{i}$

At any conjugate point $(x_{1},\dots,x_{n})$, we have $q(x)=0$, so:

$-x_{1}^{k}=x_{1}^{k-1}l_{1}(x_{2},\dots,x_{n})+\dots+l_{k}(x_{2},\dots,x_{n})\\\
$

and therefore

$|x_{1}|^{k}<|x_{1}|^{k-1}|l_{1}|+\dots+|l_{k}|$

Combining this and (4.4.3), we get an inequality for $|x_{1}|$ at any
conjugate point $(x_{1},\dots,x_{n})$, where the constant $C$ ultimately
depends on bounds for the first few derivatives of $F$:

(4.4.4)
$|x_{1}|^{k}<C\max\\{|x_{1}|,\dots,|x_{n}|\\}^{k-1}\max\\{|x_{2}|,\dots,|x_{n}|\\}$

If $|x_{1}|>\max\\{|x_{2}|,\dots,|x_{n}|\\}$, then
$|x_{1}|^{k}<C|x_{1}|^{k-1}\max\\{|x_{2}|,\dots,|x_{n}|\\}$. If the opposite
holds, then $|x_{1}|^{k}<C\max\\{|x_{2}|,\dots,|x_{n}|\\}^{k}$. So we get:

$|x_{1}|<\max\\{C,1\\}\max\\{|x_{2}|,\dots,|x_{n}|\\}$

This is the statement that all conjugate points near $x$ lie in a cone of
fixed width containing the hyperplane $x_{1}=0$. Thus all functions
$\lambda_{j}$ to $\lambda_{j+k}$ are Lipschitz at $(x_{2}\dots,x_{n})$ with a
constant independent of $x$. ∎

###### Remark.

A proof of lemma 4.2.5 in the language of section 4.3.1 seems possible: let
$\Lambda(M)$ be the bundle of Lagrangian submanifolds of the symplectic linear
spaces $T^{\ast}_{p}M$ and let $\Sigma(M)$ be the union of the Maslov cycles
within each $\Lambda_{p}(M)$. Define $\lambda:V\rightarrow\Lambda(M)$ where
$\lambda(x)$ is the tangent to $\Theta$ at $D(\Phi(x))$ (recall (4.3.2)). The
graphs of the functions $\lambda_{k}$ are the preimage of the Maslov cycle
$\Sigma(M)$. The geodesic vector field (transported to $T^{\ast}M$), is
transversal to the Maslov cycle. Showing that the angle (in an arbitrary
metric) between this vector field and the Maslov cycle at points of
intersection can be bounded from below is equivalent to showing that the
$\lambda_{k}$ are Lipschitz.

######  Lemma 4.4.1.

For any split locus $S$ and point $y\in\partial M$, there are no conjugate
points in the curve $t\rightarrow\exp(ty)$ for $t<\rho_{S}(y)$. In other
words, $\rho_{S}\leq\lambda_{1}$.

###### Proof.

Assume there is $x$ with $\rho_{S}(x)-\varepsilon>\lambda_{1}(x)$. By [Wa,
3.4], the map $F$ is not injective in any neighborhood of
$(x,\lambda_{1}(x))$. There are points $F(x_{n},t_{n})$ of $S$ with
$x_{n}\rightarrow x$ and $t_{n}<\rho_{S}(x)-\varepsilon$ (otherwise $S$ does
not split $M$). Taking limits, we see $F(x,t)$ is in $S$ for some
$t<\rho_{S}(x)-\varepsilon$, which contradicts the definition of
$\rho_{S}(x)$. ∎

From now on and for the rest of the paper, $S$ will always be a balanced split
locus:

######  Lemma 4.4.2.

Let $E\subset\partial M$ be an open subset whose closure is compact and has a
neighborhood where $\rho<\lambda_{1}$. Then $\rho_{S}$ is Lipschitz in $E$.

###### Proof.

The map $G(x)$ defined in 1.2.1, written as $(F(x),dF_{x}(r))$ is an embedding
of $V$ into $TM$. There is a constant $c$ such that for $x,y\in V$:

(4.4.5) $|F(x)-F(y)|+|dF_{x}(r)-dF_{y}(r)|\geq c\min\\{|x-y|,1\\}$

Recall the exponential map is a local diffeomorphism before the first
conjugate point. Points $p=F((z,\rho(z)))$ for $z\in E$ have a set $R_{p}$
consisting of the vector $dF_{(z,\rho(z))}(r)$, and vectors coming from