M_g,n ¯M_g,n

#  $C^{2}$-Morse functions on the Deligne-Mumford compactification of the
moduli space $\mathcal{M}_{g,n}$

Changjie Chen

###### Abstract.

We construct a series of $C^{2}$-Morse functions on the Deligne-Mumford
compactification $\overline{\mathcal{M}}_{g,n}$ of the moduli space of genus
$g$ Riemann/hyperbolic surfaces with $n$ punctures. This series of functions
converges to the systole function, which is topologically Morse. We will show
that the critical points of our functions approach those of systole
sublinearly, stratum-wise, and with the same indices. These functions might be
the first explicit examples of $C^{2}$-Morse functions on
$\overline{\mathcal{M}}_{g,n}$, and the Morse handle decomposition may give
rise to the first example of a natural cell decomposition of
$\overline{\mathcal{M}}_{g,n}$, that works no matter if $n$ is positive or
equal to 0.

Brown University, 151 Thayer St, Providence, RI, United States 02910
Email<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and A Few Theorems
    1. 2.1 Moduli space and Teichmüller space
    2. 2.2 Weil-Petersson metric
    3. 2.3 Fenchel-Nielson coordinates
    4. 2.4 Geodesic-length function
    5. 2.5 Alternative definitions
  3. 3 The systole function and Akrout’s theorem
    1. 3.1 The systole function
    2. 3.2 Topological Morse function
    3. 3.3 Akrout’s Eutacticity
    4. 3.4 Akrout’s theorem
  4. 4 The $\operatorname{sys_{T}}$ Functions and Fan Decomposition
  5. 5 Behavior of $\operatorname{sys_{T}}$ in $\mgn$
    1. 5.1 In the Major Subspace
    2. 5.2 Existence of Critical Points Near Eutactic Points
    3. 5.3 Nondegeneracy and Local Uniqueness of Critical Points
    4. 5.4 Regular Points in $\mgn$
    5. 5.5 Near the Boundary
  6. 6 Extension onto the Boundary
  7. 7 Weil-Petersson Gradient Flow

##  1\. Introduction

Morse functions are important and interesting to construct and study on
special spaces, as one can develop the Morse homology of the base space
through the handle decomposition, see [8] for more about Morse theory. The
moduli space $\mgn$ of genus $g$ hyperbolic or Riemann surfaces with $n$
punctures, as well as its Deligne-Mumford compactification $\mgnb$, is of
utmost interest while the topology is still mysterious for general $(g,n)$.
Morse theory could be a powerful tool if a Morse function is well understood
on either space.

Note that in the most popular version, a Morse function is required to be
smooth, but to define critical points and nondegeneracy, and to construct the
handle decomposition and perform Morse theory, it is allowable to weaken the
smoothness condition to $C^{2}$. We may call them $C^{2}$-Morse functions. In
this paper, we will construct and study a series of $(C^{2}\text{-})$Morse
functions using Teichmüller theory, namely the $\operatorname{sys_{T}}$
functions.

Geodesic-length functions play a fundamental role in this paper. Note that any
geodesic-length function $l_{\gamma}$ is trivially a Morse function on the
Teichmüller space. However, to descend to the moduli space, it is not possible
to take the arithmetic average over the mapping class group orbit of $\gamma$
because of the size.

While not smooth, topological Morse property has been shown for the systole
function, i.e., the minimum of all geodesic-length functions. It is not hard
to see that the systole function depends only on the hyperbolic structure of
the base surface and therefore is mapping class group invariant and descends
to the moduli space. However, as a minimum function, it is only continuous.
Explicit examples with Fenchel-Nielsen coordinates can show it is not even
$C^{1}$.

P. Schmutz Schaller showed in [13] that when $n>0$, the systole function is
topologically Morse on $\mgn$, which is an analogy of (smooth) Morse functions
in topological settings, see [9]. The case of $(g,n)=(1,1)$ has been well
studied, for instance, see [2]. Schmutz Schaller also calculated all the
critical points and the index at each of them for
$(g,n)=(0,5),(0,6),(1,2),(2,0)$ in addition to $(0,4)$ and $(1,1)$.

The general case was proved by H. Akrout in [3] a few years later: the systole
function is topologically Morse on the moduli space of a hyperbolic surface
with or without punctures, namely no matter when $n>0$ or $n=0$. Akrout
defined a minimal gradient at a surface $X\in\mathcal{T}$ to be the gradient
vector of the geodesic-length function for a shortest geodesic on $X$ and
developed a theory on (semi-)eutacticity (see Section 3 for definitions)
differentiating surfaces by the relative position of all minimal gradients for
the proof.

Aside from the stories originating from geodesic-length functions, as another
candidate, $-\log\det^{*}\Delta_{m}$ on the space of hyperbolic structures
$\\{m\\}$ on a given surface was conjectured by Sarnak to be a Morse function,
where $\det^{*}\Delta_{m}$ is the zeta regularized product of all non-zero
eigenvalues of $\Delta_{m}$, see [11]. It seems that very little study
followed and this conjecture still remains wide open.

Now we introduce a series of functions as follows, that converges to the
systole function and will have (at least) $C^{2}$-Morse property and more:

######  Definition.

For a hyperbolic surface $X$, with the parameter $T>0$,

$\operatorname{sys_{T}}(X)=-T\log\sum_{\gamma\text{ s.c.g. on
}X}e^{-\frac{1}{T}l_{\gamma}(X)},$

where s.c.g. stands for simple closed geodesic.

We will show:

######  Main Theorem.

As $T>0$ decreases to 0, $\operatorname{sys_{T}}$ increases and converges to
$\operatorname{sys}$. Moreover, for all sufficiently small $T$,
$\operatorname{sys_{T}}$ has the following properties:
(1) Every $\operatorname{sys_{T}}$ is a $C^{2}$-Morse function on the Deligne-
Mumford compactification $\mgnb$ (with altered differential structure).
(2) The Weil-Petersson gradient flow of $\operatorname{sys_{T}}$ on $\mgnb$ is
well defined.
(3) $\operatorname{Crit}(\operatorname{sys_{T}}:\mgnb\to\mathbb{R})$ with
$\operatorname{ind}(\operatorname{sys_{T}})$ respects the stratification: let
$\mathcal{S}\subset\mgnb$ be a stratum that is isomorphic to
$\mathcal{M}_{g^{\prime},n^{\prime}}$, then under the isomorphism,

$\operatorname{Crit}(\operatorname{sys_{T}}:\mathcal{S}\to\mathbb{R})\text{
and
}\operatorname{Crit}(\operatorname{sys_{T}}:\mathcal{M}_{g^{\prime},n^{\prime}}\to\mathbb{R})$

are the same, counted with index.
(4) There is a natural stratum-wise correspondence:

$\operatorname{Crit}(\operatorname{sys_{T}})\leftrightarrow\operatorname{Crit}(\operatorname{sys}).$

More precisely, let $\mathcal{S}\subset\mgnb$ be a stratum that is isomorphic
to $\mathcal{M}_{g^{\prime},n^{\prime}}$, then there is a bijection

$\displaystyle\operatorname{Crit}(\operatorname{sys_{T}}|_{\mathcal{S}})$
$\displaystyle\leftrightarrow\operatorname{Crit}(\operatorname{sys}|_{\mathcal{M}_{g^{\prime},n^{\prime}}})$
$\displaystyle p_{T}$ $\displaystyle\leftrightarrow p$

with the properties

$d_{\text{WP}}(p,p_{T})<CT,$

which implies $p_{T}\to p$ and consequently
$\operatorname{Crit}(\operatorname{sys_{T}}|_{\mathcal{S}})\to\operatorname{Crit}(\operatorname{sys}|_{\mathcal{M}_{g^{\prime},n^{\prime}}})$,
and

$\operatorname{ind}_{\operatorname{sys_{T}}}(p_{T})=\operatorname{ind}_{\operatorname{sys}}(p).$

######  Remark 1.1.

Let $f$ be a smooth function on $\mathbb{R}$ that is equal to 1 on
$[0,\operatorname{sys}(g,n)]$ and is supported on its 1-neighborhood, where
$\operatorname{sys}(g,n)=\max_{X\in\mathcal{T}}\\{\operatorname{sys}(X)\\}$,
then the function

$\operatorname{sys}_{T}^{f}(X)=-T\log\sum_{\gamma\text{ s.c.g. on
}X}f(l_{\gamma}(X))e^{-\frac{1}{T}l_{\gamma}(X)}$

is smooth Morse on $\mgnb$ for sufficiently small $T$ and has the same
properties as in the main theorem. The author thanks Rich Schwartz for this
idea.

The author believes that the $\operatorname{sys_{T}}$ functions are smooth. To
the best of the author’s knowledge, they are the first explicit examples in
literature, of $C^{2}$-Morse functions on the moduli space as well as the
Deligne-Mumford compactification. With the Morse property, they can be used to
study the topology of $\mgn$ or $\mgnb$. They can also be used to study the
systole function itself given the index-preserving critical point attracting
property.

There has been discussion and construction of cell decompositions of the
moduli space $\mgn$, when $n>0$, using quadratic differentials or fat graphs,
for example, [1] and [10]. However, it does not seem that there is a natural
cell decomposition constructed of $\mgn$ or the Deligne-Mumford
compactification $\mgnb$ for general $(g,n)$, i.e., no matter if $n$ is
positive or equal to 0. The Morse handle decomposition of $\mgnb$ given by the
$\operatorname{sys_{T}}$ functions gives rise to a cell decomposition.

As the Morse property of $\operatorname{sys_{T}}$ functions is established,
many related questions can be asked, for example, about the distribution of
critical points. In [6], the author proves a nonexistence theorem of low index
critical points for $\operatorname{sys_{T}}$ (and $\operatorname{sys}$) in
large $\mgn$, namely, all low index critical points for
$\operatorname{sys_{T}}$ live in the Deligne-Mumford boundary $\partial\mgn$,
as well as a classification of all low index critical points in all $\mgn$. As
a result, all low degree handles in the Morse handle decomposition of $\mgnb$
live in a neighborhood of the boundary. This result will lead to a method to
calculate $H^{1}(\mgn)$ or $H^{1}(\mgnb)$ by calculating some combinatorics.

Figure 1. Organization of the paper by types of points in $\mgnb$

Here is the organization of this paper. In Section 2, we review some basics in
hyperbolic geometry and Teichmüller theory and list some powerful theorems
that we will use in our later proofs. In Section 3, we talk about Akrout’s
(semi-)eutacticity theory and his theorem on the systole function, and we
start to study $\operatorname{sys_{T}}$ and develop general knowledge
including the fan decomposition to study the critical points. Subsection 5.1
is focused on the behavior of $\operatorname{sys_{T}}$ on the major subspace
while Subsection 5.2 reveals its full local behavior near eutactic points.
Subsection 5.3 establishes nondegeneracy of $\operatorname{sys_{T}}$ at its
critical points. The final pieces of the puzzle to show that
$\operatorname{sys_{T}}$ is a Morse function on $\mgn$ come in Subsection 5.4
with a study of its behavior near semi-eutactic and biased points, and
Subsection 5.5 for points near the Deligne-Mumford boundary. The extension of
$\operatorname{sys_{T}}$ to the boundary is defined and studied in Section 6
to complete the proof of the main theorem. We show a picture of the Weil-
Petersson gradient flow of $\operatorname{sys_{T}}$ on
$\overline{\mathcal{M}}_{1,1}$ in Section 7.

A table of commonly used notations in this paper is attached at the end.

##  2\. Preliminaries and A Few Theorems

###  2.1. Moduli space and Teichmüller space

Let $X$ be a (smooth) surface with negative Euler characteristic and
$\mathcal{M}_{-1}(X)$ the set of all finite area hyperbolic metrics on $X$.
Let $\textit{Diff}_{+}(X)$ be the set of all orientation-preserving
diffeomorphisms and $\textit{Diff}_{0}(X)\subset\textit{Diff}_{+}(X)$ the
subset consisting of those homotopic to the identity map, and they act on
$\mathcal{M}_{-1}(X)$ by pullback. The quotient
$\textit{Diff}_{+}(X)/\textit{Diff}_{0}(X)$ is known as the mapping class
group $\textit{Mod}(X)$. The moduli space is

$\mathcal{M}(X):=\mathcal{M}_{-1}(X)/\textit{Diff}_{+}(X).$

and the Teichmüller space is

$\mathcal{T}(X):=\mathcal{M}_{-1}(X)/\textit{Diff}_{0}(X).$

Consequently,

$\mathcal{M}(X)=\mathcal{T}(X)/\textit{Mod}(X).$

We will frequently use a hyperbolic surface $X$ instead of a hyperbolic metric
$m$ on the surface $X$, by abuse of notation, to refer to a point in the
Teichmüller space or moduli space. For diffeomorphic surfaces
$X_{1}\stackrel{{\scriptstyle f}}{{\cong}}X_{2}$, the moduli spaces
$\mathcal{M}(X_{1})$ and $\mathcal{M}(X_{2})$ are isomorphic by pullback of
$f$. When $X$ is of genus $g$ and with $n$ punctures, we also use
$\mathcal{T}_{g,n}$ and $\mgn$ for $\mathcal{T}(X)$ and $\mathcal{M}(X)$
respectively.

###  2.2. Weil-Petersson metric

The $L^{2}$ metric on $\mathcal{M}_{-1}$ at $m=Edx^{2}+2Fdxdy+Gdy^{2}$ is
defined as

$\left\langle\sigma_{1},\sigma_{2}\right\rangle_{L^{2}}=\int_{X}\frac{A_{1}A_{2}+2B_{1}B_{2}+C_{1}C_{2}}{EG-F^{2}}dxdy,$

for $\sigma_{i}=A_{i}dx^{2}+2B_{i}dxdy+C_{i}dy^{2}\in
T_{m}\mathcal{M}_{-1},i=1,2$.

The $L^{2}$ metric descends onto the Teichmüller space and the moduli space,
in which cases is known as the Weil-Petersson metric.

###  2.3. Fenchel-Nielson coordinates

Suppose $X$ is a genus $g$ hyperbolic surface with $n$ punctures. One can find
exactly $3g-3+n$ mutually disjoint simple closed geodesics on $X$ and there
are no more simple closed geodesic disjoint from them. Cutting $X$ along these
geodesics will yield some three times holed/punctured spheres, which are
called pants. A pair of pants is uniquely determined by the three cuff
lengths. A pants decomposition of $X$ is any such maximal collection of
disjoint simple closed geodesics on $X$.

Given a pants decomposition $\\{\gamma_{1},\cdots,\gamma_{3g-3+n}\\}$ of $X$,
for every $\gamma_{i}$, one can assign two numbers, $l_{i}\in\mathbb{R}_{+}$,
known as its length parameter, and $\tau_{i}\in\mathbb{R}$, , known as its
twist parameter. For any such $(l_{i},\tau_{i})_{i}$, one can construct a new
surface by cutting $X$ along all $\gamma_{i}$, changing the length of each
$\gamma_{i}$ to $l_{i}$ and gluing back $X$ along each $\gamma_{i}$ after
twisting $\tau_{i}$. This surgery gives a bijective map between
$(\mathbb{R}_{+}\times\mathbb{R})^{3g-3+n}$ to $\mathcal{T}(X)$, and
$(l_{i},\tau_{i})_{i}$ is known as the Fenchel-Nielson coordinates associated
to this pants decomposition.

###  2.4. Geodesic-length function

Given any homotopy class $\gamma$ on $X$, there exists a unique geodesic
representative with respect to a hyperbolic metric $m$, which is the unique
minimizer of the length of a curve in $\gamma$. Let the geodesic-length
function $l_{\gamma}$ associated to $\gamma$ at $(X,m)$ be the length of the
geodesic representative.

######  Theorem 2.1 (Wolpert [14]).

Any geodesic-length function $l_{\gamma}$ is real analytic on $\mathcal{T}$
and strictly convex along Weil-Petersson geodesics.

Let $s_{X}(L)$ be the number of simple closed geodesics with length at most
$L$ on a hyperbolic surface $X$. The following theorem gives an upper bound of
the growth rate of length functions.

######  Theorem 2.2 (Birman-Series [4]).

There exists a universal polynomial $P$ such that $s_{X}(L)\leq P(L)$.

We may suppose $P(x)\leq ce^{x}$ for some fixed constant $c$ for convenience.

Below we cite a few results due to Wolpert on the Weil-Petersson gradient
vector of geodesic-length functions. The main tool used in proof was the
explicit Weil-Petersson geodesic length-length formula given by Riera in [12].

######  Theorem 2.3 (Wolpert [15] 3.11, 3.16).

There exists a universal constant $c>0$, such that

$\left\langle\nabla l_{\gamma},\nabla
l_{\gamma}\right\rangle_{\textit{WP}}\leq
c\left(l_{\gamma}+l_{\gamma}^{2}e^{\frac{l_{\gamma}}{2}}\right)$

and

$\nabla^{2}l_{\gamma}\left(\cdot,\cdot\right)<c\left(1+l_{\gamma}e^{\frac{l_{\gamma}}{2}}\right)\left\langle\cdot,\cdot\right\rangle_{\textit{WP}},$

for any simple closed geodesic $\gamma$, where $\nabla^{2}$ is the Weil-
Petersson Hessian.

######  Theorem 2.4 (Wolpert [15] 3.12).

Let $\gamma_{i},\gamma_{j}$ be disjoint or identical geodesics, then

$0<\left\langle\nabla l_{i},\nabla
l_{j}\right\rangle_{\textit{WP}}=\frac{2}{\pi}l_{i}\delta_{ij}+O(l_{i}^{2}l_{j}^{2}),$

where the right hand side is uniform for $l_{i},l_{j}\leq c_{0}$.

To bound the size of vectors, we may use expressions similar to
$v(t)=u(t)+O(f(t))$ as $t\to t_{0}$, which means $\|v(t)-u(t)\|=O(f(t))$ as
$t\to t_{0}$ when a metric or norm is specified.

Let $J$ be the Weil-Petersson almost complex structure. Near the boundary of
the moduli space where mutually disjoint geodesics $\beta_{i}$’s are short,
let $\gamma_{i}$’s be a few geodesics disjoint from $\cup\beta_{i}$, such that
$\\{\lambda_{i}:=\nabla l_{\beta_{i}}^{\frac{1}{2}},J\lambda_{i},\nabla
l_{i}\\}$ is a local frame, then under this assumption:

######  Theorem 2.5 (Wolpert [16]).

Let $D$ be the Weil-Petersson connection, then as $l_{\beta_{i}}\to 0$,

$\displaystyle D_{J\lambda_{i}}\lambda_{i}=\frac{3}{2\pi
l_{\beta_{i}}^{\frac{1}{2}}}J\lambda_{i}+O(l_{\beta_{i}}^{\frac{3}{2}}),$
$\displaystyle D_{\lambda_{j}}\lambda_{i}=O(l_{\beta_{i}}^{\frac{3}{2}}),$
$\displaystyle
D_{J\lambda_{j}}\lambda_{i}=O(l_{\beta_{i}}(l_{\beta_{j}}^{\frac{3}{2}}+l_{\beta_{i}}^{\frac{1}{2}})),\text{when
}i\neq j,$ $\displaystyle D_{\nabla l_{j}}\lambda_{i}=O(l_{\beta_{i}}),$
$\displaystyle D_{\lambda_{j}}\nabla l_{i}=O(l_{\beta_{j}}^{\frac{1}{2}}),$
$\displaystyle D_{J\lambda_{j}}\nabla l_{i}=O(l_{\beta_{j}}^{\frac{1}{2}}),$
$\displaystyle D_{\nabla l_{j}}\nabla l_{i}\text{ is continuous}.$

###  2.5. Alternative definitions

We also mention some equivalent definitions commonly seen in literature.

Marking. A marking of a hyperbolic surface (Riemann surface) $X$ is a pair
$(Y,f)$ of a hyperbolic surface (Riemann surface) $Y$ and an orientation-
preserving diffeomorphism $f:X\to Y$. Two markings $(Y_{1},f_{1})$ and
$(Y_{2},f_{2})$ are equivalent if and only if $f_{2}\circ f_{1}^{-1}$ is
isotopic to an isometry (biholomorphism). The quotient $\\{(Y,f)\\}/\sim$ can
be defined as the Teichmüller space $\mathcal{T}(X)$.

Beltrami differential. A Beltrami differential on $X$ is a differential form
of type (1,-1), and on a local chart can be written as
$f\frac{d\overline{z}}{dz}$. The $L^{\infty}$-norm of such a Beltrami
differential $\mu$ is $\left\lVert\mu\right\rVert_{\infty}=\sup_{z\in
X}|f(z)|$.

For any map $f$ from $X$ to a surface $Y$, the Beltrami coefficient of $f$ is
$\mu_{f}=\frac{f_{\overline{z}}}{f_{z}}\frac{d\overline{z}}{dz}$. Specially,
if $f$ is an orientation-preserving diffeomorphism, we have
$\left\lVert\mu_{f}\right\rVert_{\infty}<1$. Let $B(X)_{1}$ be the set of
Beltrami differentials of norm $<1$ and a Beltrami coefficient is an element
in $B(X)_{1}$. In fact, the Teichmüller space can be identified with
$B(X)_{1}/\operatorname{Diff}_{0}(X)$, where $\operatorname{Diff}(X)$ acts on
$B(X)_{1}$ by pullback.

A harmonic Beltrami differential is a Beltrami differential that equals
$\omega^{-1}\overline{\varphi}$, for a holomorphic quadratic differential
$\varphi$ on $X$, where $\omega$ is the area form of the hyperbolic metric on
$X$. Let $\textit{HB}(X)$ be the space of harmonic Beltrami differentials.
Following Bers’ embedding and Bers’ projection, we have the identification of
the tangent space of the Teichmüller space

$T_{X}\mathcal{T}(X)\cong\textit{HB}(X).$

Weil-Petersson metric. The Weil-Petersson metric on $\mathcal{T}(X)$, under
the identification, is

$\left\langle\mu_{1},\mu_{2}\right\rangle_{\textit{WP}}=\int_{X}\mu_{1}\overline{\mu}_{2}\omega,$

for $\mu_{1},\mu_{2}\in\textit{HB}(X)\cong T_{m}\mathcal{T}(X)$.

The Weil-Petersson metric descends to the moduli space since it is mapping
class group invariant. See [7] for more details on Teichmüller theory.

##  3\. The systole function and Akrout’s theorem

###  3.1. The systole function

The systole function sys assigns a surface the length of a shortest
(nontrivial) closed geodesic, i.e.,

$\operatorname{sys}(X)=\min_{\gamma\text{ closed geodesic on
}X}l_{\gamma}(X).$

As the function respects isometries, it descends to
sys$:\mathcal{T}\to\mathbb{R}_{+}$, as well as
sys$:\mathcal{M}\to\mathbb{R}_{+}$.

######  Definition 3.1.

With the same assumptions on $X$ above, let $S(X)$ be the set of all shortest
closed geodesics on it.

For any $\gamma\in S(X)$, note that $\gamma$ has to be simple (otherwise it is
possible to take a nontrivial shortcut). Therefore, it is true that

$\operatorname{sys}(X)=\min_{\gamma\text{ simple closed geodesic on
}X}l_{\gamma}(X).$

It is not hard to see that $\operatorname{sys}$ is continuous but not $C^{1}$.
Examples can be constructed with Fenchel-Nielsen coordinates for instance.

###  3.2. Topological Morse function

To show Akrout’s result on $\operatorname{sys}$, we review some definitions.

######  Definition 3.2 (Topological Morse function).

Let $M^{n}$ be an $n$-dimensional manifold and $f:M\to\mathbb{R}$ a continuous
function.
(1) A point $x\in M$ is called ($C^{0}$-)regular if under a $C^{0}$-chart near
$x$, $f$ is a coordinate function, otherwise it is called ($C^{0}$-) critical.
(2) A critical point $x$ is nondegenerate if there is a local $C^{0}$-chart
$(x^{i})$ such that

$f-f(x)=(x^{1})^{2}+\cdots+(x^{r})^{2}-(x^{r+1})^{2}-\cdots(x^{n})^{2}.$

In this case the index of $f$ at $x$ is defined to be $n-r$.
(3) A continuous function is called topologically Morse if all the critical
points are nondegenerate.

For more studies, see [9].

###  3.3. Akrout’s Eutacticity

######  Definition 3.3.

Let $\\{v_{i}\\}\subset\mathbb{R}^{n}$ be a finite set of nonzero vectors. It
is called eutactic (semi-eutactic) if the origin is contained in the interior
(boundary) of the convex hull of $\\{v_{i}\\}$, with respect to the topology
of $\operatorname{Span}\\{v_{i}\\}$. We may call it biased if it is neither
eutactic or semi-eutactic.

Figure 2. In this example, $\\{v_{1},v_{2}\\}$ is biased,
$\\{v_{3},v_{4},v_{5}\\}$ is eutactic, and
$\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ is semi-eutactic.

Fix a metric, we have the following equivalent definitions:

######  Lemma 3.4.

Let $\\{v_{i}\\}\subset\mathbb{R}^{n}$ be a finite subset, then
(1) $\\{v_{i}\\}$ is eutactic if $\max_{\tau\in
S^{n-1}}\min_{i}\\{\left\langle v_{i},\tau\right\rangle\\}<0$.
(2) $\\{v_{i}\\}$ is semi-eutactic if $\max_{\tau\in
S^{n-1}}\min_{i}\\{\left\langle v_{i},\tau\right\rangle\\}=0$.
(3) $\\{v_{i}\\}$ is biased if $\max_{\tau\in S^{n-1}}\min_{i}\\{\left\langle
v_{i},\tau\right\rangle\\}>0$.

######  Definition 3.5.

Let $S(X)=\\{\gamma_{1},\cdots,\gamma_{r}\\}$ be the set of all shortest
geodesics on $X$, then the gradient vectors $\nabla l_{1},\cdots,\nabla
l_{r}\in T_{X}\mathcal{T}$ are called the minimal gradients for $X$.

We extend the concept of (semi-)eutacticity to hyperbolic surfaces.

######  Definition 3.6.

A hyperbolic surface $X\in\mathcal{T}$ is called eutactic (semi-eutactic,
biased) if the set of minimal gradients $\\{\nabla l_{i}\\}_{\gamma_{i}\in
S(X)}$ is eutactic (semi-eutactic, biased) in the tangent space
$T_{X}\mathcal{T}$ (equivalently $\\{dl_{i}\\}_{\gamma_{i}\in S(X)}$ is
eutactic (semi-eutactic, biased) in the cotangent space $T^{*}_{X}\mathcal{T}$
by duality).

###  3.4. Akrout’s theorem

In [3], Akrout proved topological Morse property for $\operatorname{sys}$
based on eutacticity:

######  Theorem 3.7.

The systole function is topologically Morse on $\mgn$ for any $(g,n)$.
$X\in\mgn$ is a critical point if and only if $X$ is eutactic, and in that
case the index is equal to $\operatorname{rank}\\{\nabla
l_{i}\\}_{\gamma_{i}\in S(X)}$.

This paper is related to but will not use Akrout’s theorem on the topological
Morse property to prove any result.

##  4\. The $\operatorname{sys_{T}}$ Functions and Fan Decomposition

In this section, we study the convergence of the $\operatorname{sys_{T}}$
functions and develop a method that the author calls fan decomposition that
will be used to study the behavior of $\operatorname{sys_{T}}$ in the
following sections.

######  Definition 4.1.

We take a family of averages of the length functions associated to all simple
closed geodesics on the base surface $X$, indexed by $T>0$:

$\operatorname{sys_{T}}(X)=-T\log\sum_{\gamma\text{ simple closed geodesic on
}X}e^{-\frac{1}{T}l_{\gamma}(X)}.$

######  Remark 4.2.

As it respects the mapping class group action, $\operatorname{sys_{T}}$
descends to $\mathcal{T}$ as well as $\mathcal{M}$. We make no distinguishment
between them by abuse of notation.

######  Definition 4.3.

Let $S(X)$ be the set of all shortest geodesics on $X$.

######  Definition 4.4.

Let $\textit{sec.sys}(X)$ be the second systole function on a hyperbolic
surface $X$, taking the value of the length of second shortest geodesics
(counted without multiplicity).

Note that the second systole function is upper semi-continuous and thus
bounded from above on the thick part of the moduli space.

######  Theorem 4.5.

The function $\operatorname{sys_{T}}$ is well defined for $T$ sufficiently
small, and it uniformly converges to $\operatorname{sys}$ as $T\to 0^{+}$.
More specifically,

$0<\operatorname{sys}(X)-\operatorname{sys_{T}}(X)<c^{\prime}T,$

where $c^{\prime}$ is a constant depending only on $(g,n)$. Moreover,
$\operatorname{sys_{T}}:\mathcal{M}\to\mathbb{R}$ is at least $C^{2}$.

###### Proof.

Write

$\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(X)}=\sum_{n=0}\sum_{n\leq
l_{\gamma}<n+1}e^{-\frac{1}{T}l_{\gamma}(X)},$

and by bounding the growth rate

$\displaystyle\sum_{n\leq l_{\gamma}<n+1}e^{-\frac{1}{T}l_{\gamma}(X)}$
$\displaystyle\leq s_{X}(n+1)e^{-\frac{1}{T}n}$ $\displaystyle\leq
ce^{n+1}e^{-\frac{1}{T}n}=ce^{(1-\frac{1}{T})n+1}.$

Therefore, $\operatorname{sys_{T}}$ is well defined for $T<1$, and

$\displaystyle\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(X)}$
$\displaystyle=re^{-\frac{1}{T}\operatorname{sys}(X)}+\sum_{n=\operatorname{sec.sys}(X)}\sum_{n\leq
l_{\gamma}<n+1}e^{-\frac{1}{T}l_{\gamma}(X)}$ $\displaystyle\leq
re^{-\frac{1}{T}\operatorname{sys}(X)}+\sum_{n=\operatorname{sec.sys}(X)}ce^{(1-\frac{1}{T})n+1}$
$\displaystyle=re^{-\frac{1}{T}\operatorname{sys}(X)}+c\frac{e^{1+\operatorname{sec.sys}(X)}}{1-e^{1-\frac{1}{T}}}e^{-\frac{1}{T}\operatorname{sec.sys}(X)},$

where $r=\\#S(X)$. Note that $r\geq 1$ is bounded by Birman-Series’ theorem as

$\operatorname{sys}(g,n):=\max_{X\in\mgn}\\{\operatorname{sys}(X)\\}<\infty$

and $\operatorname{sec.sys}$ is also bounded from above, then

$\displaystyle|\operatorname{sys_{T}}(X)-\operatorname{sys}(X)|<c^{\prime}T.$

The first and second derivatives will be studied in Lemma 4.9 and Lemma 5.13,
and that will complete the proof that $\operatorname{sys_{T}}$ is $C^{2}$. ∎

######  Lemma 4.6.

$\operatorname{sys_{T}}$ is decreasing in $T$.

###### Proof.

Note that

$\displaystyle\frac{d}{dT}\operatorname{sys_{T}}$
$\displaystyle=-\log\left(\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}\right)-\frac{1}{T}\frac{\sum_{\gamma}le^{-\frac{1}{T}l_{\gamma}}}{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}}$
$\displaystyle=-\frac{1}{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}}\left(\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}\log\left(\sum_{\gamma^{\prime}}e^{-\frac{1}{T}l}\right)-\sum_{\gamma}\left(-\frac{1}{T}l_{\gamma}\right)e^{-\frac{1}{T}l_{\gamma}}\right)$
$\displaystyle=-\frac{1}{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}}\left(\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}\left(\log\left(\sum_{\gamma^{\prime}}e^{-\frac{1}{T}l_{\gamma^{\prime}}}\right)-\left(-\frac{1}{T}l_{\gamma}\right)\right)\right)<0.$

∎

Given $T$, the first derivatives can be calculated directly by the chain rule

$\nabla\operatorname{sys_{T}}(X)=\frac{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(X)}\nabla
l_{\gamma}(X)}{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(X)}}.$

Therefore, to study the critical points of $\operatorname{sys_{T}}$, it
suffices to study

$\sum_{\gamma}e^{-\frac{1}{T}(l_{\gamma}(X)-\operatorname{sys}(X_{0}))}\nabla\operatorname{sys_{T}}(X)=\sum_{\gamma}e^{-\frac{1}{T}(l_{\gamma}(X)-\operatorname{sys}(X_{0}))}\nabla
l_{\gamma}(X)$

instead for a fixed $X_{0}$.

######  Definition 4.7.

(1) Let $p$ be a critical point of the systole function, and let

$\tilde{\Omega}_{T}(X)=\sum_{\gamma}e^{-\frac{1}{T}(l_{\gamma}(X)-\operatorname{sys}(p))}\nabla
l_{\gamma}(X),$

which is a rescaling of $\nabla\operatorname{sys_{T}}(X)$. We leave out $p$ in
the definition of $\tilde{\Omega}$ for simplicity as there are only finitely
many such critical points and the context will be clear when $\tilde{\Omega}$
is used.

(2) Let $u:(-a,a)\to\mathcal{T}$ be a unit speed geodesic with $u(0)=p$ and
tangent vector $\tau:=u^{\prime}(0)$. We let

$\tilde{\Omega}_{T}(v)=\tilde{\Omega}_{T}(t,\tau)=\tilde{\Omega}_{T}(u(t))$

by abuse of notation, where $X=u(t)$ and $v=t\tau$, where we omit the
exponential map $\operatorname{exp}$ at $p$.

(3) Let $\Omega_{T}$ be the ‘main part’ of $\tilde{\Omega}_{T}$, where we
replace the sum $\sum_{\gamma}$ which is over all simple closed geodesics by
$\sum_{\gamma\in S(p)}$ over all shortest geodesics. In the rest of the paper,
we treat other maps like $\Phi_{T}$ and $\Psi_{T}$ in the same manner. We may
also use a lowercase superscript $i$ for the $i$-th component of a sum, for
example,

$\Omega^{i}(X)=e^{-\frac{1}{T}(l_{i}(X)-\operatorname{sys}(p))}\nabla
l_{i}(X),$

and an uppercase superscript $J$ for the partial sum over $J$, for example,

$\Omega^{J}(X)=\sum_{j\in
J}e^{-\frac{1}{T}(l_{j}(X)-\operatorname{sys}(p))}\nabla l_{j}(X).$

(4) Let $\pi:\mathbb{R}^{n}\setminus\\{O\\}\to S^{n-1}$ be the standard
projection onto the unit sphere.

######  Remark 4.8.

Note that if we choose an orthonormal basis for $T_{p}\mathcal{T}$, then by
composing the exponential map $\operatorname{exp}_{p}$, we get a normal
coordinate system locally near $p$. Therefore, we can see $\tilde{\Omega}_{T}$
as a vector field on $\mathcal{T}$ near $p$ or $T_{p}\mathcal{T}$ near 0. We
will recall this later. We will focus on the renormalized gradient vector
$\tilde{\Omega}_{T}$.

######  Lemma 4.9.

When $T<1$,

$\left\lVert\tilde{\Omega}_{T}(X)-\Omega_{T}(X)\right\rVert<c^{\prime\prime}e^{-\frac{1}{T}\operatorname{sec.sys}(X)},$

where $c^{\prime\prime}$ is a constant depending on $(g,n)$.

###### Proof.

We have the growth rate of the first derivative

$\displaystyle\frac{1}{e^{-\frac{1}{T}\operatorname{sys_{T}}}}\left\lVert\sum_{n\leq
l_{\gamma}<n+1}e^{-\frac{1}{T}l_{\gamma}(X)}\nabla l_{\gamma}\right\rVert$
$\displaystyle\leq\frac{1}{e^{-\frac{1}{T}(\operatorname{sys}+c^{\prime}T)}}s_{X}(n+1)e^{-\frac{1}{T}n}\left\lVert\nabla
l_{\gamma}\right\rVert$ $\displaystyle\leq
Cce^{n+1}e^{-\frac{1}{T}n(c(n+1+(n+1)^{2}e^{\frac{n+1}{2}}))^{\frac{1}{2}}}$
$\displaystyle\leq
2Cc^{\frac{3}{2}}e^{(\frac{5}{4}-\frac{1}{T})n+\frac{5}{4}}.$

Therefore, $\operatorname{sys_{T}}$ is at least $C^{1}$. For the approximation
with $\Omega_{T}$, with the help of Theorem 2.3, we have

$\displaystyle\left\lVert\tilde{\Omega}_{T}(X)-\Omega_{T}(X)\right\rVert$
$\displaystyle\leq\sum_{\gamma\not\in
S(X)}e^{-\frac{1}{T}l_{\gamma}(X)}\left\lVert\nabla l_{\gamma}(X)\right\rVert$
$\displaystyle=\sum_{n=\operatorname{sec.sys}(X)}\sum_{n\leq
l_{\gamma}<n+1}e^{-\frac{1}{T}l_{\gamma}(X)}\|\nabla l_{\gamma}(X)\|$
$\displaystyle\leq\sum_{n=\operatorname{sec.sys}(X)}2c\sqrt{c_{1}}e^{n+1}e^{-\frac{1}{T}l_{\gamma}(X)}e^{1+\frac{n+1}{4}}$
$\displaystyle\leq
2c\sqrt{c_{1}}\frac{e^{\frac{5}{4}+\frac{9}{4}\operatorname{sec.sys}(X)}}{1-e^{\frac{9}{4}-\frac{1}{T}}}e^{-\frac{1}{T}\operatorname{sec.sys}(X)}$
$\displaystyle\leq c^{\prime\prime}e^{-\frac{1}{T}\operatorname{sec.sys}(X)}$

The lemma follows. ∎

######  Definition 4.10.

Let $X$ be a point in the Teichmüller space, and
$S(X)=\\{\gamma_{1},\cdots,\gamma_{r}\\}$ the set of all shortest geodesics on
it. Define subspaces of the tangent space $T_{X}\mathcal{T}$:

$\text{the\ }\textit{major
subspace}:T_{X}^{\operatorname{sys}}\mathcal{T}=\operatorname{Span}\\{\nabla
l_{1},\cdots,\nabla l_{r}\\},$ $\text{and\ the\ }\textit{minor
subspace}:T_{X}^{\operatorname{sys}\perp}\mathcal{T}=(T_{X}^{\operatorname{sys}}\mathcal{T})^{\perp}.$

The notions are used mainly when $X$ is eutactic.

######  Definition 4.11 (Fan decomposition of the major subspace
$T_{p}^{\operatorname{sys}}\mathcal{T}$).

Let $p\in\mathcal{M}$ be a critical point for the systole function, and
$S(p)=\\{\gamma_{i}\\}_{i\in I}$ where $I=\\{1,\cdots,r\\}$ is the index set,
then the minimal gradient set $\\{\nabla l_{i}\\}_{i\in I}$ satisfies the
eutactic condition at $p$, say $\sum_{i=1}^{r}a_{i}\nabla l_{i}(p)=0$ for
positive $a_{i}$’s. Let $J$ be a subset of $I$, and we define $F_{J}\subset
T_{p}^{\operatorname{sys}}\mathcal{T}$ to be the region consisting of vectors
$v$ such that $J$ is the set of all indices $j$ maximizing $\left\langle\nabla
l_{j},v\right\rangle$. On the other hand, for any $\tau$ it assigns a
multiindex $J=J(\tau)$ by inclusion $\tau\in F_{J}$. We extend the definition
of $J(\cdot)$ to
$\left(T_{X}^{\operatorname{sys}\perp}\mathcal{T}\right)^{c}$, the complement
of the major subspace, by pre-composing the projection,
$J=J\circ\operatorname{proj}_{T_{p}^{\operatorname{sys}}\mathcal{T}}$, by
abuse of notation. Let $v_{J}=\pi\left(\sum_{j\in J}\nabla l_{j}\right)$ be
the unitized average.

Figure 3. Fan decomposition about
$\\{v_{1},v_{2},v_{3},v_{4}\\}\subset\mathbb{R}^{2}$

The fan decomposition concerns the variation of the length of the shortest
geodesics at $p$ when the base point is perturbed. Fix the direction $\tau$ of
a small perturbation, the geodesics indexed by $J(\tau)$ will be longer than
others. If we perturb the base point in the opposite direction $-\tau$, then
the geodesics indexed by $J$ will be shorter than others and are the only
candidates for the shortest geodesics on $u(-t\tau)$, i.e.
$S(u(-t\tau))\subset\\{\gamma_{j}\\}_{J}$.

######  Lemma 4.12.

The following are true:
(1) $\bigcup_{J}F_{J}=T_{p}^{\operatorname{sys}}\mathcal{T}$.
(2) Any $F_{J}$ is a polygonal cone properly contained in some half space of
the major subspace and $\nabla l_{j}\in\mathbb{H}(\tau)$ for any $\tau$ and
all $j\in J(\tau)$, and therefore $v_{J}$ is well defined.
(3) $F_{J}\subset\mathbb{H}(v_{J})$, equivalently, $\left\langle
v_{J},\tau\right\rangle>0$ if $\tau\in F_{J}$.
(4) There exists $D>0$ such that $|\left\langle\nabla
l_{j},\tau\right\rangle|>D$ for any $\tau$ unit and all $j\in J(\tau)$, i.e.,

$\max_{\tau;j\in J(\tau)}\left\langle\nabla l_{j},\tau\right\rangle>D\text{
and }\min_{\tau;j\in J(\tau)}\left\langle\nabla l_{j},\tau\right\rangle<-D.$

(5) The inner products $\left\langle\nabla l_{j},\tau\right\rangle$ cannot all
be equal and

$\\{i:\left\langle\nabla l_{i},\tau\right\rangle\geq 0\\}\neq\emptyset\text{
and }\\{i:\left\langle\nabla l_{i},\tau\right\rangle<0\\}\neq\emptyset.$

###### Proof.

(1) is tautological. For any $0\neq v\in
T_{p}^{\operatorname{sys}}\mathcal{T}$, by the eutactic condition 3.6, there
exists at least one $i$ such that $\left\langle\nabla l_{i},v\right\rangle>0$,
otherwise, $v$ would live in the minor subspace
$T_{p}^{\operatorname{sys}\perp}\mathcal{T}$, from which (2) and (5) follows.
(3) follows directly from (2). (4) follows from the finiteness of
$\\{F_{J}\\}$ and the first two propositions. ∎

######  Remark 4.13.

If $q\in\mathcal{T}$ is semi-eutactic, we may do a fan decomposition on the
subspace spanned by the maximal eutactic subset of all minimal gradients in a
similar way, or a degenerated fan decomposition on the subspace spanned by all
minimal gradients. In the latter case, we can similarly establish an
inequality resembling (4) above:
There exists $D>0$ such that

$\max\left\langle\nabla l_{j},\tau\right\rangle>D\text{ or
}\min\left\langle\nabla l_{j},\tau\right\rangle<-D$

for any $\tau\in T^{\operatorname{sys}}\mathcal{T}$ unit.

######  Remark 4.14.

Property (4) in Lemma 4.12 and the remark above still hold true (with possibly
smaller $D$), if we allow $\tau$ to be any vector in $T_{*}\mathcal{T}$ with
$\angle(\tau,T_{*}^{\operatorname{sys}\perp}\mathcal{T})>\theta$ for fixed
$\theta>0$.

##  5\. Behavior of $\operatorname{sys_{T}}$ in $\mgn$

We review a classic exercise in topology.

######  Lemma 5.1.

Let $f,g:S^{n}\to S^{n}$ be two self maps where $f(x)+g(x)\neq 0$, then $\deg
f=\deg g$. Specially, if $g=\textit{id}$, then $\deg f=1$.

###### Proof.

$H(x,t)=\frac{(1-t)f(x)+tg(x)}{\|(1-t)f(x)+tg(x)\|}$ is a homotopy from $f$ to
$g$. ∎

###  5.1. In the Major Subspace

This subsection provides an insight of the behavior of
$\operatorname{sys_{T}}$ (though not complete), in a simplified setting, to
help one better understand, while very little from this subsection will be
directly used later.

We take the following approximation of $\Omega_{T}$:

$\Phi_{T}(t,\tau)=\sum_{i}e^{-\frac{t}{T}\left\langle\nabla
l_{i}(p),\tau\right\rangle}\nabla l_{i}(p).$

Note that rigorously we are considering the vector field
$\operatorname{exp}_{p}^{*}\Omega_{T}$ on the tangent bundle
$T(T_{p}\mathcal{T})\cong T_{p}\mathcal{T}$, as $\operatorname{exp}_{p}$ is
locally an isomorphism but we will continue to use $\Omega_{T}$ for simplicity
by abuse of notation (we will do the same thing a few times in this paper),
and the approximation $\Phi_{T}$ is done in the same space with the help of
normal coordinates.

######  Lemma 5.2.

Under the notations in Definition 4.7 and Definition 4.10,
$\Phi_{T}(\cdot,\cdot)$ restricted as a map:
$T_{p}^{\operatorname{sys}}\mathcal{T}\to
T_{p}^{\operatorname{sys}}\mathcal{T}$ has a zero.

###### Proof.

Consider $\Phi_{T}(t,-\tau)$ restricted onto the $\rho T$-sphere
$S^{d-1}_{\rho T}:=\\{t\tau:t=\rho T,\tau\text{\ unit}\\}\subset
T_{p}^{\operatorname{sys}}\mathcal{T}$, where $d=\dim
T_{p}^{\operatorname{sys}}\mathcal{T}$ is the dimension of the major subspace
and $\rho>0$ is a constant depending only on $p$ and it will be determined
later.

For any $\tau\in T_{p}^{\operatorname{sys}}\mathcal{T}$, fix a $j\in J(\tau)$,
we then have

$\displaystyle\Phi_{T}(t,-\tau)$
$\displaystyle=\sum_{i}e^{\frac{t}{T}\left\langle\nabla
l_{i}(p),\tau\right\rangle}\nabla l_{i}(p)$
$\displaystyle=e^{\frac{t}{T}\left\langle\nabla
l_{j}(p),\tau\right\rangle}\sum_{i}e^{\frac{t}{T}(\left\langle\nabla
l_{i}(p),\tau\right\rangle-\left\langle\nabla
l_{j}(p),\tau\right\rangle)}\nabla l_{i}(p).$

Therefore, by (2) in Lemma 4.12,

$\Phi_{T}(t,-\tau)\sim e^{\rho\left\langle\nabla
l_{j}(p),\tau\right\rangle}\sum_{i\in J}\nabla l_{i}(p)\neq 0,$

when $\rho=\frac{t}{T}$ is sufficiently large and hence it induces a self map
on $S^{d-1}\subset T_{p}^{\operatorname{sys}}\mathcal{T}$ by post-composing
the projection $\pi$. It can be formulated as follows:

$\displaystyle\pi\circ\Phi_{T}(\rho,-\tau)|_{\tau\in S^{d-1}}$
$\displaystyle=\pi\circ\Phi_{T}(t,-\tau)|_{t\tau\in S^{d-1}_{\rho T}}$
$\displaystyle=\pi\left(e^{\rho\left\langle\nabla
l_{j}(p),\tau\right\rangle}\sum_{i}e^{\rho(\left\langle\nabla
l_{i}(p),\tau\right\rangle-\left\langle\nabla
l_{j}(p),\tau\right\rangle)}\nabla l_{i}(p)\right)$ $\displaystyle\to
v_{J}\text{\ as\ }\rho\to\infty.$

Therefore, there exists $\rho_{J}>0$ such that

$\angle\left(\Phi_{T}(t,-\tau),\tau\right)<\frac{\pi}{2}$

when $t>\rho_{J}T$, by (3) in Lemma 4.12. By finiteness of $\\{J\\}$, take
$\rho>\max_{J}\\{\rho_{J}\\}$ and the angle condition is then satisfied for
all $\tau\in S^{d-1}\subset T_{p}^{\operatorname{sys}}\mathcal{T}$. By Lemma
5.1, $\Phi_{T}\left(\cdot,-\cdot\right)|_{S^{d-1}_{\rho T}}$ has degree 1, and
equivalently, if we don’t negate $\tau$,

$\deg\left(\pi\circ\Phi_{T}|_{S^{d-1}_{\rho T}}\right)=(-1)^{d}.$

Therefore, $\Phi_{T}$ has a zero in the interior of the $\rho T$-sphere. ∎

The following lemma establishes uniqueness, which still holds true for
$\tilde{\Omega}_{T}$ as we will see later, while the linearity below is only
for $\Phi_{T}$.

######  Lemma 5.3.

The zero in the lemma above is unique and it linearly approaches $p$ in $T$.

###### Proof.

Uniqueness: Suppose $v_{1}\neq v_{2}$ are two zeros for $\Phi_{T}(v)$.
Consider the function
$f(s):=\left\langle\Phi_{T}((1-s)v_{1}+sv_{2}),v_{1}-v_{2}\right\rangle$ which
vanishes at $0$ and $1$. Note that $f$ is strictly increasing since each
summand of

$f(s)=\sum_{i}e^{-\frac{1}{T}\left\langle\nabla
l_{i},v_{1}\right\rangle}e^{\frac{s}{T}\left\langle\nabla
l_{i},v_{1}-v_{2}\right\rangle}\left\langle\nabla
l_{i},v_{1}-v_{2}\right\rangle$

is non-decreasing and at least one of them is strictly increasing, leading to
a contradiction.
Linearity: Suppose $(t_{0},\tau_{0})$ is the zero for $\Phi_{T_{0}}$, then
$\Phi_{T}\left(\frac{t_{0}}{T_{0}}T,\tau_{0}\right)=\Phi_{T_{0}}(t_{0},\tau_{0})=0$,
i.e., $\left(\frac{t_{0}}{T_{0}}T,\tau_{0}\right)$ is the zero for
$\Phi_{T_{0}}$. ∎

Putting Lemma 5.2 and Lemma 5.3 together, we have:

######  Theorem 5.4.

$\Phi_{T}$ restricted on $T_{p}^{\operatorname{sys}}\mathcal{T}$ has a unique
zero, and it linearly approaches $p$ in $T$.

###  5.2. Existence of Critical Points Near Eutactic Points

Recall that we suppose $p$ is a critical point for $\operatorname{sys}$, i.e.,
$p$ is eutactic. While the linear approximation $\Phi_{T}$ restricted on the
major subspace $T_{p}^{\operatorname{sys}}\mathcal{T}$ has a unique zero, it
is not strong enough to draw the same conclusion for
$\nabla\operatorname{sys_{T}}$ when the orthogonal complement
$T_{p}^{\operatorname{sys}\perp}\mathcal{T}$ is involved.

Note that the $t$ component in the coordinates $(t,\tau)$ that we are using
for a small neighborhood of $p$ is the radial coordinate. As a reminder, we
omit the exponential map ‘$\operatorname{exp}_{p}^{*}$’ in
$\operatorname{exp}_{p}^{*}l_{i}$, $\operatorname{exp}_{p}^{*}\nabla l_{i}$,
and $\operatorname{exp}_{p}^{*}\Omega_{T}$, etc., by abuse of notation, while
we use the $(t,\tau)$-coordinates. Since $l_{i}$ is real analytic, we can
write

$l_{i}(t,\tau)=l_{i}(p)+t\left\langle\nabla
l_{i}(p),\tau\right\rangle+\frac{1}{2}t^{2}\nabla^{2}l_{i}(p)(\tau,\tau)+O_{i}(t^{3})$

and

$\nabla l_{i}(t,\tau)=\nabla
l_{i}(p)+t\nabla^{2}l_{i}(p)(\tau,\cdot)+O_{i}(t^{2}).$

We take the following second order approximation with respect to $T$ as
follows for $t=\rho T$, i.e., when $t\tau\in S^{n-1}_{\rho T}$,

$\displaystyle e^{-\frac{1}{T}(l_{i}(t,\tau)-\operatorname{sys}(p))}\nabla
l_{i}(t,\tau)$
$\displaystyle=e^{-\frac{1}{T}(l_{i}(t,\tau)-\operatorname{sys}(p))}(\nabla
l_{i}(p)+\rho T\nabla^{2}l_{i}(p)(\tau,\cdot)+O(T^{2}))$
$\displaystyle=e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle+\frac{1}{2}\rho^{2}T\nabla^{2}l_{i}(p)(\tau,\tau)+O(T^{2})}\nabla
l_{i}(p)+\rho Te^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle+O(T)}\nabla^{2}l_{i}(p)(\tau,\cdot)$
$\displaystyle+e^{-\frac{1}{T}(l_{i}(t,\tau)-\operatorname{sys}(p))}O(T^{2})$
$\displaystyle=e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle+\frac{1}{2}\rho^{2}T\nabla^{2}l_{i}(p)(\tau,\tau)}\nabla
l_{i}(p)+\rho Te^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}\nabla^{2}l_{i}(p)(\tau,\cdot)+O(T^{2})$

Therefore, if we write

$\displaystyle\Omega_{T}(t,\tau)$
$\displaystyle=\sum_{i}e^{-\frac{1}{T}(t\left\langle\nabla
l_{i}(p),\tau\right\rangle+\frac{1}{2}t^{2}\nabla^{2}l_{i}(\tau,\tau))}\nabla
l_{i}(p)$ $\displaystyle+t\sum_{i}e^{-\frac{1}{T}(t\left\langle\nabla
l_{i}(p),\tau\right\rangle}\nabla^{2}l_{i}(p)(\tau,\cdot)+\epsilon_{2}(T,t,\tau),$

then $\epsilon_{2}(T,t,\tau)=O(T^{2})$ when $t=\rho T$.

######  Definition 5.5.

With the notations above, we denote the ‘main part’ by

$\Psi_{T}:=\Omega_{T}-\epsilon_{2}.$

As we observed above, $\epsilon_{2}(T,t,\tau)|_{t=\rho T}=O(T^{2})$.

We fix a constant $\theta_{0}$ below that will stay fixed and be used almost
everywhere throughout the rest of the paper:

Figure 4. The angle between the two vectors is at most
$\frac{\pi}{2}-2\theta_{0}$

######  Definition 5.6.

Let $0<\theta_{0}<\frac{\pi}{6}$ be a constant (that depends on the base
critical point $p$) that satisfies

$\angle(\nabla^{2}l_{i}(\tau,\cdot),\tau)<\frac{\pi}{2}-2\theta_{0}$

for all $i$. Existence of $\theta_{0}$ is due to
$\nabla^{2}l_{i}(\tau,\tau)>0$ by convexity in Theorem 2.1, compactness of the
unit sphere and finiteness of the number of shortest geodesics on $p$.

Recall that $d:=\dim T_{p}^{\operatorname{sys}}\mathcal{T}$ is the dimension
of the major subspace.

######  Theorem 5.7.

There exists $\rho_{0}>0$ and $\epsilon_{0}=\epsilon_{0}(\rho)>0$ such that
$\pi\circ\Psi_{T}(\cdot,\cdot)$ restricted on $S^{n-1}_{\rho T}\subset
T_{p}\mathcal{T}$ has degree $(-1)^{d}$ for $\rho>\rho_{0}$ and
$T<\epsilon_{0}$.

###### Proof.

Let $i$ be the automorphism given by

$i(v_{1}+v_{2})=v_{1}-v_{2},$

where $v_{1}\in T_{X}^{\operatorname{sys}}\mathcal{T}$ and $v_{2}\in
T_{X}^{\operatorname{sys}\perp}\mathcal{T}$. For example,
$i|_{T_{X}^{\operatorname{sys}}\mathcal{T}}=\textit{id}|_{T_{X}^{\operatorname{sys}}\mathcal{T}}$.
To show the conclusion on $\pi\circ\Psi_{T}(\cdot,\cdot)$, we consider
$i\circ\pi\circ\Psi_{T}(\cdot,-\cdot)$.

When $t=\rho T$,

$\Psi_{T}(t,-\tau)|_{S^{n-1}_{\rho T}}=\sum_{i}e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}\left(e^{-\frac{1}{2}\rho^{2}T\nabla^{2}l_{i}(p)(\tau,\tau)}\nabla
l_{i}(p)-\rho T\nabla^{2}l_{i}(p)(\tau,\cdot)\right).$

Let

$\Psi_{1}(t,-\tau)=\sum_{i}e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}e^{-\frac{1}{2}\rho^{2}T\nabla^{2}l_{i}(p)(\tau,\tau)}\nabla
l_{i}(p)$

and

$\Psi_{2}(t,-\tau)=-\sum_{i}e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}\rho T\nabla^{2}l_{i}(p)(\tau,\cdot).$

We consider the following two cases by whether $\tau$ is almost perpendicular
to the major subspace as shown in Figure 5.

Figure 5. Two positions of $\tau$: almost perpendicular to the major subspace
or not

Case 1: When
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\leq\theta_{0}$. Let
$\tau^{\prime}$ be the projection of $\tau$ onto
$T_{p}^{\operatorname{sys}\perp}\mathcal{T}$, so
$\angle(\tau^{\prime},\tau)\leq\theta_{0}$. Recall that we have
$\angle(\nabla^{2}l_{i}(p)(\tau,\cdot),\tau)<\frac{\pi}{2}-2\theta_{0}$ by the
choice of $\theta_{0}$, and then it follows that
$\angle(-\Psi_{2}(t,-\tau),\tau)<\frac{\pi}{2}-2\theta_{0}$. By triangle
inequality, we have

$\displaystyle\angle(-\Psi_{2}(t,-\tau),\tau^{\prime})$
$\displaystyle<\angle(-\Psi_{2}(t,-\tau),\tau)+\angle(\tau,\tau^{\prime})$
$\displaystyle<\frac{\pi}{2}-2\theta_{0}+\theta_{0}=\frac{\pi}{2}-\theta_{0},$

which in other words implies
$i(\Psi_{2}(t,-\tau))\in\mathbb{H}(\tau^{\prime})$ by definition of $i$. Note
that $\Psi_{1}\in
T_{p}^{\operatorname{sys}}\mathcal{T}\subset\partial\mathbb{H}(\tau^{\prime})$,
thus

$i(\Psi_{T}(t,-\tau))=\Psi_{1}(t,-\tau)+i(\Psi_{2}(t,-\tau))\in\mathbb{H}(\tau^{\prime}).$

Therefore,

$\angle(i(\Psi_{T}(t,-\tau)),\tau)<\angle(\Psi_{1}+i(\Psi_{2}),\tau^{\prime})+\angle(\tau^{\prime},\tau)<\frac{\pi}{2}+\theta_{0}.$

Case 2: When
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\geq\theta_{0}$.
Similar to what we have in (4) in Lemma 4.12, by compactness of the region of
such $\tau$’s in this case, there exists $D>0$ such that $\left\langle\nabla
l_{j}(p),\tau\right\rangle>D$ for any $\tau$ and all $j\in J(\tau)$. Let
$M,m>0$ be the maximum and minimum of $\\{\|\nabla
l_{i}(p)\|,\|\nabla^{2}l_{i}(p)(\tau,\cdot)\|\\}_{i,\tau}$ respectively, where
positivity is guaranteed by Theorem 2.1 again.

Let $K=\\{k:\left\langle\nabla l_{k},\tau\right\rangle\leq 0\\}$ that is
disjoint from $J(\tau)$. For any $j\in J(\tau)$ and $k\in K$, we have the
following estimates:

$\displaystyle\left\lVert\Psi_{1}^{j}(t,-\tau)\right\rVert$
$\displaystyle>\left\langle\Psi_{1}^{j}(t,-\tau),\tau\right\rangle$
$\displaystyle=e^{\rho\left\langle\nabla
l_{j}(p),\tau\right\rangle}e^{-\frac{1}{2}\rho^{2}T\nabla^{2}l_{j}(p)(\tau,\tau)}\left\langle\nabla
l_{j}(p),\tau\right\rangle$ $\displaystyle>e^{\rho
D}e^{-\frac{1}{2}\rho^{2}TM}D,$
$\displaystyle\left\lVert\Psi_{1}^{k}(t,-\tau)\right\rVert$
$\displaystyle=e^{\rho\left\langle\nabla
l_{k}(p),\tau\right\rangle}e^{-\frac{1}{2}\rho^{2}T\nabla^{2}l_{k}(p)(\tau,\tau)}\left\lVert\nabla
l_{k}(p)\right\rVert$ $\displaystyle\leq 2e^{\frac{1}{2}\rho^{2}TM}M,$

and

$\displaystyle\left\lVert\Psi_{2}(t,-\tau)\right\rVert$
$\displaystyle\leq\sum_{i}e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}\rho
T\left\lVert\nabla^{2}l_{i}(p)(\tau,\cdot)\right\rVert$ $\displaystyle\leq
re^{\rho M}\rho TM,$

where $r=\\#S(p)$.

Note that all $\left\langle\Psi_{1}^{i}(t,-\tau),\tau\right\rangle\geq 0$ for
$i\in(J(\tau)\cup K)^{c}$, then putting all terms together we have

$\displaystyle\left\langle\Psi_{T}(t,-\tau)|_{t=\rho T},\tau\right\rangle$
$\displaystyle=\left\langle\Psi_{1}^{J}+\Psi_{1}^{K}+\Psi_{1}^{(J\cup
K)^{c}}+\Psi_{2},\tau\right\rangle$
$\displaystyle\geq\left\langle\Psi_{1}^{J},\tau\right\rangle-\left\lVert\Psi_{1}^{K}\right\rVert-\left\lVert\Psi_{2}\right\rVert$
$\displaystyle\geq e^{\rho
D}e^{-\frac{1}{2}\rho^{2}TM}D-2e^{\frac{1}{2}\rho^{2}TM}M-re^{\rho M}\rho TM,$

where $J=J(\tau)$ for short. If we set $T<e^{-\rho M}$, the expression above
will go to $+\infty$ as $\rho\to\infty$. Therefore, we fix $\rho$ such that

$\left\langle\Psi_{T}(t,-\tau)|_{t=\rho T},\tau\right\rangle>0$

for any $T<e^{-\rho M}$.

With the choice of $\rho$ and the two cases considered above,

$\deg i\circ\pi\circ\Psi_{T}(\cdot,-\cdot)=1.$

The theorem follows. ∎

To make the approximation effective as in Theorem 5.9, we bound the size of
$\Psi_{T}$.

######  Lemma 5.8.

With the same notations above, we have the following estimates on the norms
when $t=\rho T$:
(1) When
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\leq\theta_{0}$,

$\left\lVert\Psi_{T}(t,-\tau)\right\rVert>\rho Te^{-\rho
M\sin\theta_{0}}m\sin\theta_{0}\cos\theta_{0},$

(2) When
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\geq\theta_{0}$,

$\left\lVert\Psi_{T}(t,-\tau)\right\rVert>\frac{1}{2}e^{\rho
D}e^{-\frac{1}{2}\rho^{2}TM}D.$

Together we may claim that $\left\lVert\Psi_{T}|_{t=\rho
T}\right\rVert>C(p)T.$

###### Proof.

In the first case, let $\tau^{\prime}$ be the projection of $\tau$ onto the
minor subspace and consider the projection of $\Psi_{T}$ onto $\tau^{\prime}$.
Note that

$\left\langle\nabla l_{i}(p),\tau\right\rangle=\|\nabla
l_{i}(p)\|\cos\angle(\nabla l_{i},\tau)>-M\sin\theta_{0}$

and as a reminder, $\theta_{0}$ is chosen such that
$\angle(\nabla^{2}l_{i}(\tau,\cdot),\tau)<\frac{\pi}{2}-2\theta_{0}$, so

$\angle(\nabla^{2}l_{i}(p)(\tau,\cdot),\tau^{\prime})<\angle(\nabla^{2}l_{i}(p)(\tau,\cdot),\tau)+\angle(\tau,\tau^{\prime})<\frac{\pi}{2}-\theta_{0}$

and

$\displaystyle\left\langle\nabla^{2}l_{i}(p)(\tau,\cdot),\tau^{\prime}\right\rangle$
$\displaystyle=\left\lVert\nabla^{2}l_{i}(p)(\tau,\cdot)\right\rVert\cdot\left\lVert\tau^{\prime}\right\rVert\cos\angle(\nabla^{2}l_{i}(p)(\tau,\cdot),\tau^{\prime})$
$\displaystyle>m\sin\theta_{0}\cos\theta_{0}.$

Therefore,

$\displaystyle\left\langle\Psi_{T},-\tau^{\prime}\right\rangle$
$\displaystyle=\sum_{i}e^{\rho\left\langle\nabla
l_{i}(p),\tau\right\rangle}\rho
T\left\langle\nabla^{2}l_{i}(p)(\tau,\cdot),\tau^{\prime}\right\rangle$
$\displaystyle>\rho Te^{-\rho M\sin\theta_{0}}m\sin\theta_{0}\cos\theta_{0}$

The first claim follows. The second follows directly from the definition of
$\rho$. ∎

What Theorem 5.7 tells us is that $\Psi_{T}$ has a zero in the interior of
$B^{n}_{\rho T}$ by Lemma 5.1. Now we are ready to prove the same conclusion
for $\tilde{\Omega}_{T}$:

######  Theorem 5.9 (Existence).

With the same $\rho$ above, $\operatorname{sys_{T}}$ has a critical point in
the $\rho T$-ball $B^{n}_{\rho T}\subset T_{p}\mathcal{T}$ when $T$ is
sufficiently small.

###### Proof.

To recall the idea, we take a two-step approximation

$\tilde{\Omega}_{T}\approx\Omega_{T}\approx\Psi_{T}.$

Now let us consider the errors.

When $T$ is sufficiently small, any point $X$ in the geodesic ball
$\operatorname{exp}_{p}(B_{\rho T})$ centered at $p$ satisfies that
$\operatorname{sec.sys}(X)>\operatorname{sec.sys}(p)$, and thus by Lemma 4.9,

$\left\lVert\tilde{\Omega}_{T}(X)-\Omega_{T}(X)\right\rVert<c^{\prime\prime}e^{-\frac{1}{T}\operatorname{sec.sys}(X)}<c^{\prime\prime}e^{-\frac{1}{T}\operatorname{sec.sys}(p)}.$

For the second approximation, since $\rho$ is already fixed as in Theorem 5.7,
by Definition 5.5,

$\left\lVert\Omega_{T}(X)-\Psi_{T}(X)\right\rVert=O(T^{2}).$

Therefore, the total error can be given by

$\left\lVert\tilde{\Omega}_{T}(X)-\Psi_{T}(X)\right\rVert=O(T^{2}).$

Since the angle between $\tau$ and $i\circ\Psi_{T}(t,-\tau)|_{S^{n-1}_{\rho
T}}$ can be bounded by $\frac{\pi}{2}+\theta_{0}$ (which is not optimal), and
$\left\lVert i\circ\Psi_{T}(t,-\tau)|_{S^{n-1}_{\rho T}}\right\rVert>C(p)T$,
by the lemma above, with the error taken into effect, the conditions in Lemma
5.1 are satisfied, and

$\deg(\tilde{\Omega}_{T})=\deg(\Psi_{T})=(-1)^{d}.$

Existence follows. ∎

######  Corollary 5.10.

Suppose uniqueness (which will be proved in Theorem 5.17) is satisfied for the
existence above. Let $p_{T}$ denote the unique critical point for
$\operatorname{sys_{T}}$ near $p$, then $d(p,p_{T})$ is sublinear in $T$,
meaning $d(p,p_{T})<CT$ for some $C$.

We also prove the nonexistence of critical points outside of the $\rho T$-ball
centered at $p$:

Figure 6. Near eutactic $p$: there is a unique critical point at most $\rho T$
away from $p$ in a fixed small geodesic ball independent of $T$

######  Theorem 5.11 (Nonexistence outside the $\rho T$-ball).

There exist $\rho_{1}>0$, $\epsilon_{1}=\epsilon_{1}(\rho)>0$ and $r_{p}>0$
for each $p$, such that there are no critical points of
$\operatorname{sys_{T}}$ with distance from $p$ in $[\rho T,r_{p}]$ for
$\rho>\rho_{1}$ and $T<\min\left\\{\epsilon_{1},\frac{r_{p}}{\rho}\right\\}$.

###### Proof.

We recall that $0<\theta_{0}<\frac{\pi}{6}$ is chosen such that for all $i$,

$\angle\left(\nabla^{2}l_{i}\left(\tau,\cdot\right),\tau\right)<\frac{\pi}{2}-2\theta_{0}.$

For $\tau\not\in T_{p}^{\operatorname{sys}\perp}\mathcal{T}$, let
$C^{\operatorname{sys}}(\tau,\theta)$ be the cone in the major subspace
centered at $\operatorname{proj}_{T_{p}^{\operatorname{sys}}\mathcal{T}}\tau$
with angle $\theta$. Note that $v\in C^{\operatorname{sys}}(\tau,\theta)\times
T_{p}^{\operatorname{sys}\perp}\mathcal{T}$ if and only if

$\angle(\operatorname{proj}_{T_{p}^{\operatorname{sys}}\mathcal{T}}v,\operatorname{proj}_{T_{p}^{\operatorname{sys}}\mathcal{T}}\tau)<\theta.$

For example, if $\tau\in T_{p}^{\operatorname{sys}}\mathcal{T}$, then
$C^{\operatorname{sys}}\left(\tau,\frac{1}{2}\pi\right)\times
T_{p}^{\operatorname{sys}\perp}\mathcal{T}=\mathbb{H}(\tau)$.

Let $U_{p}=B(p,r_{p})$ be a geodesic ball centered at $p$ and $C>0$ a constant
with $Cr_{p}<\frac{1}{2}$, such that for any $q\in\overline{U}_{p}$, if we let
$t=d(p,q)$ and $\tau=\frac{1}{t}\operatorname{exp}_{p}^{-1}(q)$, the following
conditions are satisfied:

(1) By continuity,

$\left|l_{i}(q)-l_{i}(p)-t\left\langle\nabla
l_{i}(p),\tau\right\rangle\right|<Ct^{2},$

for all $i$ and

$l_{j}(q)-l_{j}(p)<0,$

for $j\in J(-\tau)$,

(2) By continuity,

$\|\nabla l_{i}(q)-\nabla l_{i}(p)\|<Ct,$

for all $i$ and consequently

$\angle(\nabla l_{i}(q),\nabla
l_{i}(p))<\frac{1}{2}\min\left\\{\theta_{0},\frac{1}{2}\pi-\theta_{1}\right\\},$

where $\theta_{1}$ is chosen in Case 2 below, and

$\frac{1}{2}<\frac{\left\lVert\nabla l_{i}(q)\right\rVert}{\left\lVert\nabla
l_{i}(p)\right\rVert}<\frac{3}{2},$

(3) By continuity,

$\left\lVert\nabla l_{i}(q)-\nabla
l_{i}(p)-t\nabla^{2}l_{i}(\tau,\cdot)\right\rVert<Ct^{2},$

and by taking projection onto the minor subspace,

$t\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\nabla^{2}l_{i}(\tau,\cdot)\right\rVert-
Ct^{2}>0,$

(4) By continuity,

$\max\\{l_{i}(q)\\}<\min_{\gamma\not\in S(p)}{l_{\gamma}(q)}.$
$\operatorname{sys}(p)+\left(r_{p}\|\nabla
l_{i}\|\sin\theta_{0}+2Cr_{p}^{2}\right)<\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}$

for all $i$.

Condition (4) guarantees that we only need to consider the sum over
$\gamma_{i}\in S(p)$ since the ‘tail’ will be dominantly smaller. Given these
constants and conditions, we again consider the following two cases:

Case 1: if
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\leq\theta_{0}$, then

$\angle(\tau,\nabla l_{i}(p))\geq\frac{\pi}{2}-\theta_{0},$

for all $i\in I$, and therefore,

$\displaystyle e^{-\frac{1}{T}l_{i}(q)}$
$\displaystyle>e^{-\frac{1}{T}\operatorname{sys}(p)}e^{-\frac{1}{T}(t\left\langle\nabla
l_{i}(p),\tau\right\rangle+Ct^{2})}$
$\displaystyle>e^{-\frac{1}{T}\operatorname{sys}(p)}e^{-\frac{1}{T}(t\|\nabla
l_{i}(p)\|\sin\theta_{0}+Ct^{2})}$
$\displaystyle>e^{-\frac{1}{T}\operatorname{sys}(p)}e^{-\frac{1}{T}(r_{p}\|\nabla
l_{i}(p)\|\sin\theta_{0}+Cr_{p}^{2})}$
$\displaystyle>e^{-\frac{1}{T}(\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}-Cr_{p}^{2})}$

Take the projection of $\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\nabla l_{i}(q)$ onto
the minor subspace, then we have, for a single $\nabla l_{i}(q)$,

$\displaystyle\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\nabla
l_{i}(q)\right\rVert$
$\displaystyle>t\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\nabla^{2}l_{i}(\tau,\cdot)\right\rVert-
Ct^{2}$ $\displaystyle\geq\rho
T\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\nabla^{2}l_{i}(\tau,\cdot)\right\rVert-
Cr_{p}\rho T=:C_{1}(p)T,$

and for the main part,

$\displaystyle\left\lVert\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert$
$\displaystyle\geq\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert$
$\displaystyle\geq\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}(\nabla
l_{i}(q))\right\rVert$
$\displaystyle>\sum_{i}e^{-\frac{1}{T}\operatorname{sys}(p)}e^{-\frac{1}{T}(r_{p}\left\lVert\nabla
l_{i}(p)\right\rVert\sin\theta_{0}+Cr_{p}^{2})}\left(t\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\nabla^{2}l_{i}(\tau,\cdot)\right\rVert-
Ct^{2}\right)$ $\displaystyle>e^{-\frac{1}{T}(\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}-Cr_{p}^{2})}C_{1}(p)T.$

We thus have the positivity of the norm of the gradient vector, as the tail is
dominantly smaller:

$\displaystyle\left\lVert\sum_{\gamma}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert$
$\displaystyle\geq\left\lVert\operatorname{proj}_{T_{p}^{\operatorname{sys}\perp}\mathcal{T}}\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert-\left\lVert\sum_{\gamma\not\in
S(p)}e^{-\frac{1}{T}l_{i}(q)}\nabla l_{i}(q)\right\rVert$
$\displaystyle>C_{1}(p)Te^{-\frac{1}{T}(\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}-Cr_{p}^{2})}-C_{2}(p)e^{-\frac{1}{T}\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}}>0.$

Case 2: if
$\angle(\tau,T_{p}^{\operatorname{sys}\perp}\mathcal{T})\geq\theta_{0}$. By
eutacticity, there exists $0<\theta_{1}<\frac{1}{2}\pi$ such that for any
$\tau\not\in T_{p}^{\operatorname{sys}\perp}\mathcal{T}$,

$C^{\operatorname{sys}}\left(\tau,\frac{1}{2}\pi-\theta_{1}\right)\cap\\{\nabla
l_{i}\\}\neq\emptyset.$

Let

$J:=\left\\{j:\nabla l_{j}(p)\in
C^{\operatorname{sys}}\left(-\tau,\frac{1}{2}\pi-\theta_{1}\right)\right\\}$

(note that $J$ is not short for $J(-\tau)$ in this case) and

$K:=\left\\{k:\nabla l_{k}(p)\in
C^{\operatorname{sys}}\left(\tau,\frac{\pi}{2}+\frac{1}{2}\theta_{1}\right)\right\\}.$

By continuity, there exist $D_{1}<0$ and $D_{2}$ constant, such that when we
can make $r_{p}$ small enough, for any $q\in U_{p}=B(p,r_{p})$:
(1) For $j\in J$,

$\left\langle\nabla l_{j}(q),\tau\right\rangle<D_{1}.$

(2) For $k\in K$,

$\left\langle\nabla l_{k}(q),\tau\right\rangle>D_{2}.$

(3) For $i\in(J\cup K)^{c}$,

$\left\langle\nabla l_{i}(q),\tau\right\rangle<0.$

(4) $D_{1}<D_{2}$.

For any $j\in J$ and $k\in K$, we have

$\displaystyle\frac{\left\lVert e^{-\frac{1}{T}l_{k}(q)}\nabla
l_{k}(q)\right\rVert}{\left\lVert e^{-\frac{1}{T}l_{j}(q)}\nabla
l_{j}(q)\right\rVert}$
$\displaystyle<\frac{\frac{3}{2}e^{-\frac{1}{T}l_{k}(q)}\left\lVert\nabla
l_{k}(p)\right\rVert}{\frac{1}{2}e^{-\frac{1}{T}l_{j}(q)}\left\lVert\nabla
l_{j}(p)\right\rVert}$
$\displaystyle=3e^{-\frac{1}{T}(l_{k}(q)-l_{j}(q))}\frac{\left\lVert\nabla
l_{k}(p)\right\rVert}{\left\lVert\nabla l_{j}(p)\right\rVert}$
$\displaystyle<3e^{\frac{1}{T}(-t(\left\langle\nabla
l_{k}(p),\tau\right\rangle-\left\langle\nabla
l_{j}(p),\tau\right\rangle)+2Ct^{2})}\frac{\|\nabla l_{k}(p)\|}{\|\nabla
l_{j}(p)\|}$
$\displaystyle<3e^{\frac{1}{T}(t(D_{1}-D_{2})+2Ct^{2})}\frac{\left\lVert\nabla
l_{k}(p)\right\rVert}{\left\lVert\nabla l_{j}(p)\right\rVert}$
$\displaystyle<3e^{\rho(D_{1}-D_{2})+2Cr_{p}}\frac{\left\lVert\nabla
l_{k}(p)\right\rVert}{\left\lVert\nabla l_{j}(p)\right\rVert}<\epsilon,$

where $\epsilon<\frac{1}{r}$, if $\rho$ is sufficiently large.

We consider the projection of $\sum_{i}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)$ onto the major subspace. For any $\nabla l_{j}(p)\in
C^{\operatorname{sys}}\left(-\tau,\frac{1}{2}\pi-\theta_{1}\right)$,

$\displaystyle-\left\langle e^{-\frac{1}{T}l_{j}(q)}\nabla
l_{j}(q),\tau\right\rangle$
$\displaystyle>-e^{-\frac{1}{T}\operatorname{sys}(p)}e^{-\frac{1}{T}(l_{j}(q)-l_{j}(p))}\left\langle\nabla
l_{j}(q),\tau\right\rangle$
$\displaystyle>-D_{1}e^{-\frac{1}{T}\operatorname{sys}(p)}e^{\frac{1}{T}(-t\left\langle\nabla
l_{j},\tau\right\rangle-Ct^{2})}$
$\displaystyle>-D_{1}e^{-\frac{1}{T}\operatorname{sys}(p)}e^{\frac{1}{T}(-D_{1}\rho
T-Cr_{p}\rho T)}$
$\displaystyle=-D_{1}e^{-\frac{1}{T}\operatorname{sys}(p)}e^{\rho(-D_{1}-Cr_{p})}$

Note that, for any $i\in(J\cup K)^{c}$, $\nabla l_{i}(q)$ has a negative
projection onto $\tau$. Therefore,

$\displaystyle\left\lVert\sum_{I}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert$
$\displaystyle>-\left\langle\sum_{K^{c}}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q),\tau\right\rangle-\left\lVert\sum_{K}e^{-\frac{1}{T}l_{k}(q)}\nabla
l_{k}(q)\right\rVert$
$\displaystyle\geq-\left\langle\sum_{J}e^{-\frac{1}{T}l_{j}(q)}\nabla
l_{j}(q),\tau\right\rangle-\left\lVert\sum_{K}e^{-\frac{1}{T}l_{k}(q)}\nabla
l_{k}(q)\right\rVert$ $\displaystyle\geq-(1-r\epsilon)\left\langle
e^{-\frac{1}{T}l_{j}(q)}\nabla l_{j}(q),\tau\right\rangle$

Put the tail in,

$\displaystyle\left\lVert\sum_{\gamma}e^{-\frac{1}{T}l_{i}(q)}\nabla
l_{i}(q)\right\rVert$ $\displaystyle>-(1-r\epsilon)\left\langle
e^{-\frac{1}{T}l_{j}(q)}\nabla
l_{j}(q),\tau\right\rangle-\left\lVert\sum_{\gamma\not\in
S(p)}e^{-\frac{1}{T}l_{i}(q)}\nabla l_{i}(q)\right\rVert$
$\displaystyle>-(1-r\epsilon)D_{1}e^{-\frac{1}{T}\operatorname{sys}(p)}e^{\rho(-D_{1}-Cr_{p})}-C_{2}(p)e^{-\frac{1}{T}\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}}$
$\displaystyle>C_{3}(p)e^{-\frac{1}{T}\operatorname{sys}(p)}-C_{2}(p)e^{-\frac{1}{T}\min_{\gamma\not\in
S(p)}{l_{\gamma}(q)}}>0$

The theorem follows. ∎

######  Remark 5.12.

Let $p_{1},p_{2}$ be two critical points. Although it seems not feasible to
calculate $r_{i},i=1,2$, but we have

$d(p_{1},p_{2})>\max\\{r_{p_{1}},r_{p_{2}}\\}.$

###  5.3. Nondegeneracy and Local Uniqueness of Critical Points

As the actual systole function is topologically Morse, we naturally hope the
$C^{2}$ functions $\operatorname{sys_{T}}$ in the approximation family are
$C^{2}$-Morse or even smooth Morse, when $T$ is sufficiently close to 0. To
proceed, with the same notations above, we directly calculate the Hessian
$\tilde{H}_{\operatorname{sys_{T}}}$ of $\operatorname{sys_{T}}$ on the
tangent vector field $\tau=u^{\prime}$ for any geodesic $u$:

$\displaystyle\tilde{H}_{T}(\tau)$
$\displaystyle:=T\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)^{2}\tilde{H}_{\operatorname{sys_{T}}}(\tau,\tau)$
$\displaystyle=\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\left\langle\nabla
l_{\gamma},\tau\right\rangle\right)^{2}-\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\left\langle\nabla
l_{\gamma},\tau\right\rangle^{2}\right)\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)$
$\displaystyle+T\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\nabla^{2}l_{\gamma}(\tau,\tau)\right)$

Let $H_{T}$ be the ‘main part’ of $\tilde{H}_{T}$, i.e., where every sum above
is over $S(p)$ instead of all simple closed geodesics. By Theorem 2.3 and a
similar calculation as in Lemma 4.9 and Theorem 5.9,

######  Lemma 5.13.

We have

$\displaystyle\left|(\tilde{H}_{T})_{p}-(H_{T})_{p}\right|<c^{\prime\prime\prime}e^{-\frac{1}{T}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}.$

###### Proof.

Note that

$\displaystyle\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\nabla^{2}l_{\gamma}(\tau,\tau)\leq
c\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}(1+l_{\gamma}e^{\frac{l_{\gamma}}{2}})\leq
c(p)e^{-\frac{1}{T}\operatorname{sec.sys}(p)},$

then the lemma follows from Theorem 4.5 and Lemma 4.9. ∎

Our hope is that $\tilde{H}_{T}$ is nondegenerate at critical points of
$\operatorname{sys_{T}}$, but first as an observation we have the following:

######  Lemma 5.14.

When $T$ is sufficiently small,
$(\tilde{H}_{T})_{p}:T_{p}\mathcal{T}\to\mathbb{R}$ is nondegenerate.

###### Proof.

Recall that $r=\\#S(X)$. Since all $\gamma_{i}$’s attain the same value at
$p$, the above can be simplified as

$(H_{T})_{p}(\tau)=-e^{-\frac{2}{T}\operatorname{sys}(p)}\left(r\sum_{i}\left<\nabla
l_{i},\tau\right>^{2}-\left(\sum_{i}\left\langle\nabla
l_{i},\tau\right\rangle\right)^{2}-Tr\sum_{i}\nabla
l_{i}^{2}(p)(\tau,\tau)\right).$

If $\tau\in T_{p}^{\operatorname{sys}\perp}\mathcal{T}$, then

$(\tilde{H}_{T})_{p}(\tau)>Tre^{-\frac{2}{T}\operatorname{sys}(p)}\sum_{i}\nabla^{2}l_{i}(p)(\tau,\tau)-C_{p}e^{-\frac{1}{T}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}>0.$

If $\tau\in T_{p}^{\operatorname{sys}}\mathcal{T}$, by part (5) of Lemma 4.12
and Cauchy’s inequality,

$r\sum_{i}\left\langle\nabla
l_{i},\tau\right\rangle^{2}-\left(\sum_{i}\left\langle\nabla
l_{i},\tau\right\rangle\right)^{2}>0.$

Similar to part (4) in Lemma 4.12, there exists $D>0$ (we may assume the same
$D$ for both Lemma 4.12 and here for simplicity), such that

$\min_{i}\\{\left\langle\nabla l_{i}(p),\tau\right\rangle\\}<-D.$

Let

$I^{+}=\\{i:\left\langle\nabla l_{i},\tau\right\rangle\geq 0\\}$

and

$I^{-}=\\{i:\left\langle\nabla l_{i},\tau\right\rangle<0\\},$

and also we let $r_{1}=\\#I^{+}$ and $r_{2}=\\#I^{-}$, then we have

$\displaystyle r\sum_{i}\left\langle\nabla
l_{i},\tau\right\rangle^{2}-\left(\sum_{i}\left\langle\nabla
l_{i},\tau\right\rangle\right)^{2}$ $\displaystyle=r\left(\sum_{i\in
I^{+}}\left\langle\nabla l_{i},\tau\right\rangle^{2}+\sum_{i\in
I^{-}}\left\langle\nabla l_{i},\tau\right\rangle^{2}\right)-\left(\sum_{i\in
I^{+}}\left\langle\nabla l_{i},\tau\right\rangle+\sum_{i\in
I^{+}}\left\langle\nabla l_{i},\tau\right\rangle\right)^{2}$
$\displaystyle=\left(r_{1}\sum_{i\in I^{+}}\left\langle\nabla
l_{i},\tau\right\rangle^{2}-\left(\sum_{i\in I^{+}}\left\langle\nabla
l_{i},\tau\right\rangle\right)^{2}\right)+\left(r_{2}\sum_{i\in
I^{-}}\left\langle\nabla l_{i},\tau\right\rangle^{2}-\left(\sum_{i\in
I^{-}}\left\langle\nabla l_{i},\tau\right\rangle\right)^{2}\right)$
$\displaystyle+r_{2}\sum_{i\in I^{+}}\left\langle\nabla
l_{i},\tau\right\rangle^{2}+r_{1}\sum_{i\in I^{-}}\left\langle\nabla
l_{i},\tau\right\rangle^{2}-2\left(\sum_{i\in I^{+}}\left\langle\nabla
l_{i},\tau\right\rangle\right)\left(\sum_{i\in I^{-}}\left\langle\nabla
l_{i},\tau\right\rangle\right)$ $\displaystyle\geq r_{2}\sum_{i\in
I^{+}}\left\langle\nabla l_{i},\tau\right\rangle^{2}+r_{1}\sum_{i\in
I^{-}}\left\langle\nabla l_{i},\tau\right\rangle^{2}-2\left(\sum_{i\in
I^{+}}\left\langle\nabla l_{i},\tau\right\rangle\right)\left(\sum_{i\in
I^{-}}\left\langle\nabla l_{i},\tau\right\rangle\right)$ $\displaystyle\geq
r_{2}D^{2}+r_{1}D^{2}+2D^{2}\geq 4D^{2}.$

Therefore, going back to $(H_{T})_{p}$, we have

$(H_{T})_{p}(\tau)\leq-e^{-\frac{2}{T}\operatorname{sys}(p)}\left(4D^{2}-Tr^{2}M\right),$

and

$\displaystyle(\tilde{H}_{T})_{p}(\tau)$
$\displaystyle<(H_{T})_{p}(\tau)+C_{p}e^{-\frac{1}{T}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}$
$\displaystyle\leq-e^{-\frac{2}{T}\operatorname{sys}(p)}\left(4D^{2}-Tr^{2}M\right)+C_{p}e^{-\frac{1}{T}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}$

which is negative when $T$ is sufficiently small as
$\operatorname{sec.sys}>\operatorname{sys}$.

Therefore, when $T$ is small enough, $H_{T}(\tau)<0$. ∎

Given a critical point $p_{T}$ of $\operatorname{sys_{T}}$, the Hessian can be
simplified as

$(\tilde{H}_{T})_{p_{T}}(\tau)=\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\left(-\left\langle\nabla
l_{\gamma},\tau\right\rangle^{2}+T\nabla^{2}l_{\gamma}(\tau,\tau)\right)\right).$

######  Theorem 5.15.

Let $p_{T}$ be a critical point (no uniqueness established yet) for
$\operatorname{sys_{T}}$ as in Theorem 5.9 when $T$ is sufficiently small,
then $(\tilde{H}_{T})_{p_{T}}:T_{p_{T}}\mathcal{T}\to\mathbb{R}$ is
nondegenerate.

###### Proof.

Let $v_{T}=\operatorname{exp}_{p}^{-1}(p_{T})\in T_{p}\mathcal{T}$, then
$d(\operatorname{exp}_{p})_{v_{T}}:T_{v_{T}}T_{p}\mathcal{T}\cong
T_{p}\mathcal{T}\to T_{p_{T}}\mathcal{T}$ is an isomorphism when $T$ is small.
By smoothness, let $C=C(p)>0$ satisfy

$\left|d(\operatorname{exp}_{p})_{v_{T}}(\nabla l_{i}(p))-\nabla
l_{i}(p_{T})\right|<Cd(p,p_{T})<C\rho T.$

We also let

$\text{tail}=\sum_{\gamma\not\in
S(p)}e^{-{\frac{1}{T}}l_{\gamma}}\left(-\left\langle\nabla
l_{\gamma},\tau\right\rangle^{2}+T\nabla^{2}l_{\gamma}(\tau,\tau)\right),$

and then $|\text{tail}|<C_{p}e^{-\frac{1}{T}\operatorname{sec.sys}(p)}$,
similar as above.

Instead of the tangent space $T_{p_{T}}\mathcal{T}$ at $p_{T}$, we consider
the push forward of the tangent subspaces
$T_{p}^{\operatorname{sys}}\mathcal{T}$ and
$T_{p}^{\operatorname{sys}\perp}\mathcal{T}$ at $p$. By continuity, results
similar to part (4) of Lemma 4.12 and the bounds of $\\{\|\nabla
l_{i}(p_{T})\|,\|\nabla^{2}l_{i}(p_{T})(\tau,\cdot)\|\\}_{i,\tau;T\leq\epsilon}$
can be established. For simplicity, by abuse of notation, we still denote the
bounds by $D,M,m$. We also assume

$l_{i}(p_{T})<\frac{1}{2}\left(\operatorname{sys}(p)+\operatorname{sec.sys}(p)\right)<\operatorname{sec.sys}(p)$

for all $\gamma_{i}\in S(p)$ by making $T$ small enough.

When $\tau\in
d\left(\operatorname{exp}_{p}\right)_{v_{T}}\left(T_{p}^{\operatorname{sys}}\mathcal{T}\right)$,

$\displaystyle\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)^{-1}\cdot(\tilde{H}_{T})_{p_{T}}(\tau)$
$\displaystyle\leq-e^{-\frac{1}{T}\frac{1}{2}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}(D^{2}-rTM)+C_{p}e^{-\frac{1}{T}\operatorname{sec.sys}(p)}<0,$

when $T$ is sufficiently small.

When $\tau\in
d\left(\operatorname{exp}_{p}\right)_{v_{T}}\left(T_{p}^{\operatorname{sys}\perp}\mathcal{T}\right)$,

$\displaystyle\left(\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}\right)^{-1}\cdot(\tilde{H}_{T})_{p_{T}}(\tau)$
$\displaystyle\geq
re^{-\frac{1}{T}\frac{1}{2}(\operatorname{sys}(p)+\operatorname{sec.sys}(p))}(-(2C\rho
T)^{2}+Tm)-C_{p}e^{-\frac{1}{T}\operatorname{sec.sys}(p)}>0,$

when $T$ is sufficiently small.

The theorem follows since $d(\operatorname{exp}_{p})_{v_{T}}$ is an
isomorphism. ∎

If we say the index of a (nondegenerate) quadratic form is the number of
negative eigenvalues, then we have shown in the proof above:

######  Corollary 5.16.

Let $p$ be a critical point for $\operatorname{sys}$, then

$\operatorname{ind}\tilde{H}_{\operatorname{sys_{T}}}(p)=\operatorname{ind}\tilde{H}_{\operatorname{sys_{T}}}(p_{T})=(-1)^{d}=\operatorname{ind}_{\operatorname{sys}}(p).$

######  Theorem 5.17 (Uniqueness within $\rho T$-ball).

When $T$ is sufficiently small, there exists a unique critical point $p_{T}$
within the $\rho T$-ball around $p$.

###### Proof.

Let $p_{i},i\in I$ be all the critical points in the interior of the $\rho
T$-sphere described as above. Consider the gradient vector field
$\nabla\operatorname{sys_{T}}$ on the interior of $S_{\rho T}$ that is nonzero
on the boundary. Note that

$\operatorname{ind}_{p_{i}}(\nabla\operatorname{sys_{T}})=\operatorname{ind}\tilde{H}_{\operatorname{sys_{T}}}(p_{i}).$

Therefore, when $T$ is sufficiently small, by Theorem 5.7, we have

$\displaystyle(-1)^{d}$
$\displaystyle=\deg(\nabla\operatorname{sys_{T}}|_{S_{\rho
T}})=\sum_{I}\operatorname{ind}_{p_{i}}\left(\nabla\operatorname{sys_{T}}\right)$
$\displaystyle=\sum_{I}\operatorname{ind}\tilde{H}_{\operatorname{sys_{T}}}(p_{i})=\\#I\cdot(-1)^{d},$

which implies $\\#I=1$, i.e., the critical point near $p$ is unique. ∎

The theorem eventually validates the definition we have used above:

######  Definition 5.18.

Suppose $T$ is sufficiently small. Let $p_{T}$ be the unique critical point
near $p$.

###  5.4. Regular Points in $\mgn$

By Akrout’s theorem, regular points for the systole function are exactly non-
eutactic points, namely, biased and semi-eutactic points. Similar results are
true for $\operatorname{sys_{T}}$.

######  Theorem 5.19 (biased points).

For any biased point $q\in\mathcal{M}$, there exist a neighborhood $V=V(q)$ of
$q$ and $T_{0}=T_{0}(q)>0$ such that any $q^{\prime}\in V$ is regular for
$\operatorname{sys_{T}}$ for $T<T_{0}$.

###### Proof.

Let $I$ be the index set for $S(q)$. Since $q$ does not satisfy the semi-
eutactic condition, there exists $\tau\in
T_{q}^{\operatorname{sys}}\mathcal{T}$ such that $\left\langle\nabla
l_{i}(q),\tau\right\rangle>0$ for all $i\in I$, equivalently,

$\nabla l_{i}(q)\in C\left(\tau,\frac{1}{2}\pi-\theta_{2}\right)$

for some $0<\theta_{2}(q)<\frac{1}{2}\pi$, where $C(\tau,\theta)$ is the cone
in the tangent space $T_{q}\mathcal{T}$ centered at $\tau$ with angle
$\theta$. Let $V$ be a neighborhood of $q$ such that for any $q^{\prime}\in
V$:
(1) $\nabla l_{i}(q^{\prime})\in
C\left(\tau,\frac{1}{2}\pi-\frac{1}{2}\theta_{2}\right)$,
(2) $\max\\{l_{i}(q^{\prime})\\}<\min_{\gamma\not\in
S(q)}\\{l_{\gamma}(q^{\prime})\\}$.

Therefore, we have

$\displaystyle\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(q^{\prime})}\|\nabla\operatorname{sys_{T}}(q^{\prime})\|=\left\|\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}(q^{\prime})}\nabla
l_{\gamma}(q^{\prime})\right\|$
$\displaystyle\geq\left\|\sum_{i}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|-\left\|\sum_{\gamma\not\in
S(q)}e^{-\frac{1}{T}l_{\gamma}(q^{\prime})}\nabla
l_{\gamma}(q^{\prime})\right\|$
$\displaystyle\geq\left\|\left\langle\sum_{i}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime}),\tau\right\rangle\right\|-\left\|\sum_{\gamma\not\in
S(q)}e^{-\frac{1}{T}l_{\gamma}(q^{\prime})}\nabla
l_{\gamma}(q^{\prime})\right\|$
$\displaystyle>\sin(\frac{1}{2}\theta_{2})\sum_{i}\left\|\nabla
l_{i}(q^{\prime})\right\|e^{-\frac{1}{T}\max\\{l_{i}(q^{\prime})\\}}-C(q)e^{-\frac{1}{T}\min_{\gamma\not\in
S(q)}\\{l_{\gamma}(q^{\prime})\\}}>0,$

for $T<T_{0}$ for appropriate $T_{0}>0$. ∎

To show the same property for semi-eutactic points, a few lemmas are needed
and we present them after the theorem.

######  Theorem 5.20 (Semi-eutactic points).

For any semi-eutactic point $q\in\mathcal{M}$, there exist a neighborhood
$V=V(q)$ of $q$ and $T_{0}=T_{0}(q)>0$ such that any $q^{\prime}\in V$ is
regular for $\operatorname{sys_{T}}$ for $T<T_{0}$.

###### Proof.

By Lemma 5.21 below, we split the index set $I$ of $S(q)$ into a maximal
eutactic index subset $I_{e}$ and the biased index complement $I_{b}$. Let
$V(q)$ be the geodesic ball $B(q,t_{0})$ for $t_{0}>0$ to be determined.
Recall that
$T^{\operatorname{sys}}_{q}\mathcal{T}=\operatorname{Span}_{I}\\{\nabla
l_{i}\\}$ (and let us use $T^{\operatorname{sys}}$ for short). We let

$T^{e}=\operatorname{Span}_{I_{e}}\\{\nabla l_{i}\\}\text{ and
}T^{e\perp}=\left(T^{e}\right)^{\perp}.$

Also recall that $\theta_{0}$ is chosen such that
$\angle\left(\nabla^{2}l_{i}(\tau,\cdot),\tau\right)<\frac{\pi}{2}-2\theta_{0}$
for all $i$.

By continuity, we choose $\theta_{3},\theta_{4}$ and $d$ small enough such
that
(1) $\theta_{3}<\theta_{0}$ and
$C_{2}\max\\{\theta_{3},C_{1}\theta_{4}\\}<\theta_{0}$, where $C_{1},C_{2}$
are fixed constants to be introduced in case 2(b),
(2) $\max_{i\in
I_{e},\angle(\tau,T^{e\perp})\leq\theta_{3}}|\left\langle\nabla
l_{i},\tau\right\rangle|<d$,
(3) If $\left\langle\nabla l_{k},\tau\right\rangle\geq-2d$ then $\angle(\nabla
l_{k},\tau)\leq\frac{\pi}{2}+\theta_{4}$ for $k\in I_{b}$,
and further conditions presented in the proof.

Case 1: When $\angle(\tau,T^{e\perp})\geq\theta_{3}$ (i.e., $\tau$ not
perpendicular to $T^{e}$ with angle bound).
By the semi-eutactic condition, since $\tau\not\in T^{e\perp}$, we have that
$\overline{\mathbb{H}(\tau)}\cap\\{\nabla l_{i}(q)\\}\neq\emptyset$ and
$\mathbb{H}(-\tau)\cap\\{\nabla l_{i}(q)\\}\neq\emptyset$. In fact, over the
compact region of candidate $\tau$’s in this case, there exists $D>0$ such
that

$\min_{i}\left\langle\nabla l_{i}(q),\tau\right\rangle<-D.$

Let

$J=\\{j:\left\langle\nabla
l_{j}(q),\tau\right\rangle=\min_{i}\left\langle\nabla
l_{i}(q),\tau\right\rangle\\}$

and

$K=\overline{\mathbb{H}(\tau)}.$

For any $q^{\prime}\in V(q)$, $j\in J$ and $k\in K$, we have

$\displaystyle\frac{\left\|e^{-\frac{1}{T}l_{k}(q^{\prime})}\nabla
l_{k}(q^{\prime})\right\|}{\left\|e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime})\right\|}$
$\displaystyle=e^{-\frac{1}{T}(l_{k}(q^{\prime})-l_{j}(q^{\prime}))}\frac{\left\|\nabla
l_{k}(q^{\prime})\right\|}{\left\|\nabla l_{j}(q^{\prime})\right\|}$
$\displaystyle=e^{\frac{1}{T}(t\left\langle\nabla
l_{j}(q),\tau\right\rangle-t\left\langle\nabla
l_{k}(q),\tau\right\rangle+O(t^{2}))}\frac{\left\|\nabla
l_{k}(q^{\prime})\right\|}{\left\|\nabla l_{j}(q^{\prime})\right\|}$
$\displaystyle<e^{-\frac{1}{2}\frac{1}{T}t_{0}D}\frac{\left\|\nabla
l_{k}(q^{\prime})\right\|}{\left\|\nabla l_{j}(q^{\prime})\right\|}<\epsilon,$

for $t_{0}$ small enough.

Pick a $j\in J$, and consider the sum over $K^{c}$,

$\displaystyle\left\|\sum_{K^{c}}e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime})\right\|$
$\displaystyle\geq-\left\langle\sum_{K^{c}}e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime}),\tau\right\rangle$
$\displaystyle\geq-\left\langle\sum_{J}e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime}),\tau\right\rangle$ $\displaystyle\geq
De^{-\frac{1}{T}l_{j}(q^{\prime})}$

Put all terms together,

$\displaystyle\left\|\sum_{S(q)}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$
$\displaystyle\geq\left\|\sum_{K^{c}}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$
$\displaystyle-\left\|\sum_{K}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$ $\displaystyle\geq
D\sum_{J}e^{-\frac{1}{T}l_{j}(q^{\prime})}-r\epsilon
e^{-\frac{1}{T}l_{j}(q^{\prime})}\left\|\nabla l_{j}(q^{\prime})\right\|$
$\displaystyle>(D-rM\epsilon)e^{-\frac{1}{T}l_{j}(q^{\prime})},$

where $M=\max_{i;q^{\prime}}\|\nabla l_{i}(q^{\prime})\|$. Note that by
setting $t_{0}$ small, the sum above is
$O(e^{-\frac{1}{T}(\operatorname{sys}(q)+\epsilon)})$, and the ‘tail’, the sum
over $S(q)^{c}$, is bounded by
$e^{-\frac{1}{T}(\operatorname{sec.sys}(q)-\epsilon)}=o(e^{-\frac{1}{T}(\operatorname{sys}(q)+\epsilon)})$.
Therefore, $\nabla\operatorname{sys_{T}}(q^{\prime})\neq 0$.

Case 2: When $\angle(\tau,T^{e\perp})\leq\theta_{3}$ (i.e., when $\tau$ almost
perpendicular to $T^{e}$).

Let

$\mathbf{C}=\cap_{k\in I_{b}}\mathbb{H}_{T^{\operatorname{sys}}}(\nabla
l_{k})=\\{\tau:\left\langle\nabla l_{k},\tau\right\rangle>0\text{\ for\ all\
}k\\}$

be the polygonal cone in $T^{\operatorname{sys}}$ consisting of vectors with
positive inner products with $\nabla l_{k}$ for all $k\in I_{b}$, which is
nonempty and convex by biased condition. Note that $\mathbf{C}\times
T^{\operatorname{sys}\perp}$ is the set of such vectors in the total tangent
space. We fix a $v\in\mathbf{C}$ and make $t$ sufficiently small so that
$\left\langle\nabla l_{k}(q^{\prime}),v\right\rangle$ is positively bounded
from below by continuity.

\- Case 2(a): When furthermore, $\angle(\nabla
l_{k},\tau)\leq\frac{\pi}{2}+\theta_{4}$ for all $k\in I_{b}$.
Let $J=\\{j\in I_{b}:\frac{\pi}{2}\leq\angle(\nabla
l_{j},\tau)\leq\frac{\pi}{2}+\theta_{4}\\}\subset I_{b}$ and
$K:=I_{b}\setminus J$ the complement.

Let $T^{e\perp\operatorname{sys}}$ be the orthogonal complement of $T^{e}$ in
$T^{\operatorname{sys}}$, then

$\left(\partial\mathbf{C}\times T^{\operatorname{sys}\perp}\right)\cap
T^{e\perp}=\left(\partial\mathbf{C}\cap
T^{e\perp\operatorname{sys}}\right)\times T^{\operatorname{sys}\perp},$

which is trivial if and only if $\partial\mathbf{C}\cap
T^{e\perp\operatorname{sys}}=0$ and $T^{\operatorname{sys}\perp}=0$ (and thus
$T^{e\perp\operatorname{sys}}=T^{e\perp}$). In this case $J=\emptyset$ if
$\theta_{3}$ and $\theta_{4}$ are small enough. Otherwise, we would have

$\angle\left(\partial\mathbf{C},T^{e\perp\operatorname{sys}}\right)\leq\angle\left(\tau,\partial\mathbf{C}\right)+\angle\left(\tau,T^{e\perp}\right)\leq
C_{1}\theta_{4}+\theta_{3}$

by Lemma 5.24. Contradiction!

If $J=\emptyset$, then $\tau\in\mathbf{C}\times T^{\operatorname{sys}\perp}$.
Note that in this situation $\mathbf{C}\cap T^{e\perp}$ is nonempty. Let
$\tau^{\prime}$ be the projection of $\tau$ onto $T^{e\perp}$, then

$\tau^{\prime}\in\left(\mathbf{C}\times T^{\operatorname{sys}\perp}\right)\cap
T^{e\perp}$

by convexity of $\mathbf{C}\times T^{\operatorname{sys}\perp}$ and
$\angle(\tau,\tau^{\prime})<\theta_{3}<\theta_{0}$.

If $J\neq\emptyset$, let $\tau^{\prime}$ be the projection of $\tau$ onto
$(\mathbf{C}\times T^{\operatorname{sys}\perp})\cap T^{e\perp}$, which is
nontrivial. By Lemma 5.24,

$\angle\left(\tau,\partial\mathbf{C}\times
T^{\operatorname{sys}\perp}\right)\leq\angle\left(\tau,\partial\mathbf{C}\right)<C_{1}\theta_{4}$

and since $\angle(\tau,T^{e\perp})\leq\theta_{3}$, by Lemma 5.23,

$\angle\left(\tau,\tau^{\prime}\right)<C_{2}\max\\{\theta_{3},C_{1}\theta_{4}\\}<\theta_{0}.$

Therefore, for $i\in I_{e}\cup J$, since $\nabla l_{i}(q)\perp\tau^{\prime}$,

$\left\langle\nabla
l_{i}(q^{\prime}),\tau^{\prime}\right\rangle=\left\langle\nabla
l_{i}(q^{\prime})-\nabla l_{i}(q),\tau^{\prime}\right\rangle=\left\langle
t\nabla^{2}l_{i}(\tau,\cdot)+O_{i}(t^{2}),\tau^{\prime}\right\rangle.$

Consider the angle, when $t$ is sufficiently small,

$\displaystyle\angle\left(t\nabla^{2}l_{i}(\tau,\cdot)+O_{i}(t^{2}),\tau^{\prime}\right)$
$\displaystyle\leq\angle\left(t\nabla^{2}l_{i}(\tau,\cdot)+O_{i}(t^{2}),\tau\right)+\angle\left(\tau,\tau^{\prime}\right)$
$\displaystyle\leq\frac{\pi}{2}-\frac{3}{2}\theta_{0}+\theta_{0}=\frac{\pi}{2}-\frac{1}{2}\theta_{0},$

Thus, $\left\langle\nabla l_{i}(q^{\prime}),\tau^{\prime}\right\rangle>0$.

For $k\in K$, note that $\left\langle\nabla
l_{k}(q),\tau^{\prime}\right\rangle>0$ since
$\tau^{\prime}\in\operatorname{int}(\mathbf{C}\times
T^{\operatorname{sys}\perp})$ if $J=\emptyset$ and
$\tau^{\prime}\in\partial\mathbf{C}\times T^{\operatorname{sys}\perp}$ if
$J\neq\emptyset$, then

$\displaystyle\left\langle\nabla l_{k}(q^{\prime}),\tau^{\prime}\right\rangle$
$\displaystyle=\left\langle\nabla
l_{k}(q),\tau^{\prime}\right\rangle+\left\langle
t\nabla^{2}l_{k}(q)(\tau,\cdot)+O_{k}(t^{2}),\tau^{\prime}\right\rangle>0$

where the second term is positive for the same reason as above.

Each term has a positive projection onto $\tau^{\prime}$ and thus the sum is
nonzero.

\- Case 2(b): When furthermore, $\min_{k\in I_{b}}\left\langle\nabla
l_{k},\tau\right\rangle\leq-2d$.

Let $J\subset I_{b}$ be the index set of the vectors that realize the minimum
above.

For $i\in I_{e}$, note that $\left\langle\nabla
l_{j},\tau\right\rangle-\left\langle\nabla
l_{i},\tau\right\rangle\leq-2d+d=-d$, then

$\displaystyle\frac{\left\|\operatorname{proj}_{v}(e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime}))\right\|}{\left\|\operatorname{proj}_{v}(e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime}))\right\|}$
$\displaystyle=\frac{e^{-\frac{1}{T}(t\left\langle\nabla
l_{i},\tau\right\rangle+O_{i}(t^{2}))}}{e^{-\frac{1}{T}(t\left\langle\nabla
l_{j},\tau\right\rangle+O_{j}(t^{2}))}}\frac{\left\|\operatorname{proj}_{v}(t\nabla^{2}l_{i}(q)(\tau,\cdot)+O_{i}(t^{2}))\right\|}{\left\lVert\operatorname{proj}_{v}(\nabla
l_{j}(q^{\prime}))\right\rVert}$
$\displaystyle=te^{\frac{1}{T}(t\left\langle\nabla
l_{j},\tau\right\rangle-t\left\langle\nabla
l_{i},\tau\right\rangle+O_{ij}(t^{2}))}\frac{\left\lVert\operatorname{proj}_{v}(\nabla^{2}l_{i}(q)(\tau,\cdot)+O_{i}(t))\right\rVert}{\left\|\operatorname{proj}_{v}(\nabla
l_{j}(q^{\prime}))\right\|}$ $\displaystyle\leq
te^{-\frac{1}{2}\frac{1}{T}td}\frac{\left\lVert\operatorname{proj}_{v}(\nabla^{2}l_{i}(q)(\tau,\cdot)+O_{ij}(t))\right\rVert}{\left\lVert\operatorname{proj}_{v}(\nabla
l_{j}(q^{\prime}))\right\rVert}<\epsilon,$

for $t$ sufficiently small.

By the choice of $v$, $\left\langle\nabla l_{k}(q^{\prime}),v\right\rangle$ is
bounded from below by a positive constant for all $k\in I_{b}$. Therefore,

$\displaystyle\left\|\sum_{I}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$
$\displaystyle\geq\|\operatorname{proj}_{v}\sum_{I_{e}\cup
I_{b}}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla l_{i}(q^{\prime})\|$
$\displaystyle\geq\left\|\operatorname{proj}_{v}\sum_{I_{b}}e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime})\right\|$
$\displaystyle-\left\|\operatorname{proj}_{v}\sum_{I_{e}}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$
$\displaystyle\geq\left\|\operatorname{proj}_{v}\sum_{J}e^{-\frac{1}{T}l_{j}(q^{\prime})}\nabla
l_{j}(q^{\prime})\right\|$
$\displaystyle-\left\|\operatorname{proj}_{v}\sum_{I_{e}}e^{-\frac{1}{T}l_{i}(q^{\prime})}\nabla
l_{i}(q^{\prime})\right\|$
$\displaystyle>(1-r\epsilon)\left\|e^{-\frac{1}{T}l_{j}(q^{\prime})}\operatorname{proj}_{v}(\nabla
l_{j}(q^{\prime}))\right\|>0$

Each case corresponds to a compact region of the unit sphere, and the process
in each case can be done continuously, hence theorem follows.

∎

######  Lemma 5.21.

Let $\\{v_{i}\\}_{i\in I}$ be a semi-eutactic set, then the index set $I$ can
be uniquely split into a maximal eutactic index subset $I_{e}$ and the biased
index complement $I_{b}$. See Figure 7.

###### Proof.

Consider the convex hull of $\\{v_{i}\\}$, then the origin is on the boundary
by definition. Let $P$ be the maximal face of the convex hull that the origin
lives on. Let $I_{e}$ be the index set of the maximal subset whose convex hull
is $P$, then any hyperplane perpendicular to $\operatorname{Span}\\{v_{i}\\}$
passing through $\\{v_{i}\\}_{I_{e}}$ separates all other vectors on one side.
∎

Figure 7. Split by eutacticity: in this example where $I=\\{1,2,3,4,5\\}$, we
have $I_{e}=\\{3,4,5\\}$ and $I_{b}=\\{1,2\\}$

######  Lemma 5.22.

Given a finite biased set of vectors $\\{v_{k}\\}\subset\mathbb{R}^{n}$, then
for any $\theta>0$ small, there exists $d>0$ such that for any unit vector
$\tau$ at least one of the following is true:
(1) $\angle(v_{k},\tau)\leq\frac{\pi}{2}+\theta$ for all $k$,
(2) $\min_{k}\left\langle v_{k},\tau\right\rangle\leq-2d.$

###### Proof.

This is by continuity. ∎

######  Lemma 5.23.

Let $V_{1},V_{2}\subset\mathbb{R}^{n}$ be two linear subspaces with nontrivial
intersection $V_{0}:=V_{1}\cap V_{2}$. Then there exists a constant $C$
depending on $\angle(V_{1},V_{2})$ such that for $\theta$ small enough, for
any $v\in\mathbb{R}^{n}$ with $\angle(v,V_{i})<\theta$ for $i=1,2$, we have
$\angle(v,V_{0})<C\theta$, i.e.,

$C(V_{1},\theta)\cap C(V_{2},\theta)\subset C(V_{0},C\theta).$

###### Proof.

It is obvious if one is a subspace of the other. Assume
$\angle(V_{1},V_{2})>0$. Let $v_{i}$ be the projection of $v$ onto $V_{i}$,
$i=0,1,2$ and $v^{\prime}_{i}=v_{i}-v_{0}$, $i=1,2$. Now by the setup
$\angle(V_{1},V_{2})=\min\\{\angle(v^{\prime}_{1},v^{\prime}_{2}),\pi-\angle(v^{\prime}_{1},v^{\prime}_{2})\\}>0$
and $\angle(v,V_{i})=\angle(v,v_{i})$ for $i=0,1,2$, thus it suffices to prove
the lemma in $\mathbb{R}^{3}$. Let $C_{i}$ be the great circle on $S^{2}$
spanned by $v_{i}$ and $v_{0}$, $i=1,2$, then $C_{1}\cap C_{2}=\pm v_{0}$, and
$\min\\{\angle(v^{\prime}_{1},v^{\prime}_{2}),\pi-\angle(v^{\prime}_{1},v^{\prime}_{2})\\}=\angle(C_{1},C_{2})$.
Therefore, there exists $C>0$ depending on $\angle(C_{1},C_{2})$ such that
$\angle(v,v_{0})=d_{S^{2}}(v,v_{0})<C\max_{i=1,2}{d_{S^{2}}(v,C_{i})}=C\max_{i=1,2}{\angle(v,v_{i})}<C\theta$
if $v$ is close enough to $C_{1}$ and $C_{2}$. ∎

Figure 8. Projectivized view of the settings in Lemma 5.23

######  Lemma 5.24.

Let $\\{v_{i}\\}_{i\in I}\subset\mathbb{R}^{n}$ be a finite biased set of
vectors, and $\mathbf{C}=\cap_{i}\mathbb{H}(v_{i})$, then there exists
$\theta$ small enough, such that for any vector $\tau\not\in\mathbf{C}$ with
$\angle(v_{i},\tau)\leq\frac{\pi}{2}+\theta$ for all $i$, there exists a
unique unit vector $u\in\partial\mathbf{C}$ such that
$\angle(u,\tau)=\min_{v\in\partial\mathbf{C}}\angle(v,\tau)$.

Furthermore, $\angle(u,\tau)<C\theta$ for constant $C$.

###### Proof.

It is clear that $v\in\mathbf{C}$ if and only if $\left\langle
v_{i},v\right\rangle\geq 0$ for all $i$, then for $v_{1},v_{2}\in\mathbf{C}$,
for $0\leq\lambda\leq 1$, $\left\langle v_{i},\lambda
v_{1}+(1-\lambda)v_{2}\right\rangle\geq 0$, which shows that $\mathbf{C}$ is
radial and convex. Note that $\mathbf{C}$ is contained in some half space. Now
consider the angle between a vector and $\tau$ as a function on
$\mathbf{C}\cap S^{n-1}$. If $\tau\not\in\mathbf{C}\cap S^{n-1}$,
$\angle(\cdot,\tau)=d_{S^{n-1}}(\cdot,\tau)$ is minimized over $\mathbf{C}\cap
S^{n-1}$ by a unique vector $v\in\partial\mathbf{C}\cap S^{n-1}$ by convexity
if $\tau$ is closed enough to $\partial\mathbf{C}$.

The second part is due to Lemma 5.23 and finiteness of faces. ∎

Figure 9. Projection onto the polygonal cone of a vector near the convex hull

######  Definition 5.25.

With the same settings in the lemma above, define

$\angle(\tau,\partial\mathbf{C})=\angle(u,\tau)\text{ and\
}\operatorname{proj}_{\mathbf{C}}\tau=\operatorname{proj}_{u}\tau.$

######  Remark 5.26.

We can reduce the condition on $\mathbf{C}$ to any radial and convex subset
for Lemma 5.24 and Definition 5.25.

######  Lemma 5.27.

Let $\mathbf{C}=\cap\mathbb{H}_{i}\neq\emptyset$ be the intersection of
finitely many half spaces. Suppose $V$ is a linear subspace that has
nontrivial intersection with $\mathbf{C}$, then for any $v\in\mathbb{R}^{n}$
and for $\theta$ small, if $\angle(v,\partial\mathbf{C})<\theta$ and
$\angle(v,V)<\theta$, then

$\angle(v,\mathbf{C}\cap V)\leq\angle(v,\partial\mathbf{C}\cap V)<C\theta.$

###### Proof.

Let $\tau^{\prime}$ be the projection of $\tau$ onto $\partial\mathbf{C}$ and
$f$ be a face of $\partial\mathbf{C}$ such that $\tau^{\prime}\in f$, then we
claim $f\cap V\neq\emptyset$ if $\theta$ is chosen small enough. Otherwise,
$\angle(f,V)\leq\angle(v,f)+\angle(v,V)<2\theta$ which is a contradiction if
$2\theta<\min_{f\in F}\angle(f,V)$, where $F=\\{f,f\text{ is a face of
}\partial\mathbf{C}\text{ and }f\cap V=\emptyset\\}$. Note that $f\cap V$ is
radial and convex, $\angle(\tau,\partial\mathbf{C}\cap V)\leq\angle(\tau,f\cap
V)\leq\angle(\tau,\operatorname{proj}_{f\cap V}\tau^{\prime})<C\theta$ by
Lemma 5.23. ∎

###  5.5. Near the Boundary

######  Definition 5.28.

The $\epsilon$-thin part and $\epsilon$-thick part of the moduli space are
defined as

$\mathcal{M}^{\leq\epsilon}=\\{X\in\mathcal{M}:\operatorname{sys}(X)\leq\epsilon\\},$
$\mathcal{M}^{\geq\epsilon}=\\{X\in\mathcal{M}:\operatorname{sys}(X)\geq\epsilon\\}.$

######  Theorem 5.29 (Points near the boundary).

There exist $\epsilon>0$ and $T_{0}>0$ such that any
$q\in\mathcal{M}^{\leq\epsilon}$ is an regular point for
$\operatorname{sys_{T}}$ for all $T<T_{0}$. In fact, for any $\beta$ of length
$\leq\epsilon$, $\left\langle\nabla
l_{\beta},\nabla\operatorname{sys_{T}}\right\rangle>0$.

###### Proof.

Let $q\in\mathcal{M}^{\leq\epsilon}$ and
$S(q)=\\{\gamma_{1},\cdots,\gamma_{r}\\}$. By collar lemma or Corollary 4.1.2
in [5], $\gamma_{i}$’s are disjoint. Let $\gamma_{I}=\cup\gamma_{i}$. We have
the Weil-Petersson pairing of $\nabla l_{i}$ with $\nabla l_{\gamma}$ for
$\gamma$ classified by the following three types:

Type 1: $\gamma=\gamma_{j}$.

$\left\langle\nabla l_{\gamma},\nabla l_{i}\right\rangle>\left\|\nabla
l_{i}\right\|\delta_{ij}.$

Type 2: $\gamma\cap\gamma_{I}=\emptyset$.

$\left\langle\nabla l_{\gamma},\nabla l_{i}\right\rangle>0.$

Type 3: $\gamma\not\subset\gamma_{I}$, $\gamma\cap\gamma_{I}\neq\emptyset$.

By collar lemma, $l_{\gamma}>x(\epsilon)$, where
$x(\epsilon)=2\operatorname{arcsinh}(\frac{1}{\sinh\frac{\epsilon}{2}})>-2\log\epsilon$.

Therefore, choose any $i$,

$\displaystyle\left\lVert\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}\nabla
l_{\gamma}\right\rVert$
$\displaystyle\geq\left\lVert\sum_{\gamma\cap\gamma_{I}=\emptyset\text{\ or\
}\gamma\subset\gamma_{I}}e^{-\frac{1}{T}l_{\gamma}}\nabla
l_{\gamma}\right\rVert-\left\lVert\sum_{\gamma\cap\gamma_{I}\neq\emptyset}e^{-\frac{1}{T}l_{\gamma}}\nabla
l_{\gamma}\right\rVert$ $\displaystyle\geq
e^{-\frac{1}{T}l_{i}}\left\lVert\nabla l_{i}\right\rVert-
Ce^{-\frac{1}{T}x(\epsilon)}$
$\displaystyle\geq\frac{1}{2}\sqrt{\epsilon}e^{-\frac{1}{T}\epsilon}-Ce^{\frac{2}{T}\log\epsilon}$

In order for the lower bound above to be positive, it suffices to make $T$
satisfy

$\displaystyle
T<\frac{-2\log\epsilon-\epsilon}{\log(2C)-\frac{1}{2}\log\epsilon}$

Note that

$\displaystyle\frac{-2\log\epsilon-\epsilon}{\log(2C)-\frac{1}{2}\log\epsilon}\to
4,$

as $\epsilon\to 0^{+}$. Therefore, when $\epsilon<\epsilon_{0}$ small enough,
a uniform upper bound $T_{0}$ for $T$ can be chosen such that for any
$q\in\mathcal{M}^{\leq\epsilon_{0}}$, $\operatorname{sys_{T}}(q)\neq 0$ for
all $T<T_{0}$.

∎

######  Remark 5.30.

By critical points attracting property, all critical points of
$\operatorname{sys_{T}}$ live in the thick part of the moduli space.

######  Remark 5.31.

The positive pairing in the theorem implies that when $T<T_{0}$,
$\nabla\operatorname{sys_{T}}$ points transversely inward to the
$\operatorname{sys}$-level sets $\mathcal{M}^{=\epsilon^{\prime}}$ in
$\mathcal{M}^{\leq\epsilon}$. Therefore,

######  Corollary 5.32.

For $\epsilon$ small enough, $\mathcal{M}^{\geq\epsilon}$ is a deformation
retract of $\mathcal{M}$.

##  6\. Extension onto the Boundary

Recall that the Deligne-Mumford boundary $\partial_{\textit{DM}}\mathcal{M}$
of the moduli space $\mathcal{M}(X)$ is the space of all nodal hyperbolic
surfaces modeled on $X$. We may use $\partial\mathcal{M}$ for simplicity. The
Deligne-Mumford compactification is the union

$\overline{\mathcal{M}}=\mathcal{M}\cup\partial\mathcal{M}.$

A stratum in the compactification is a maximal connected subset in which
points are homeomorphic to each other.

Let $X\in\partial\mathcal{M}$ with the pinched curve set $S$ and call a
neighborhood stratifically closed if the pinched curve set of any point is a
subset of $S$. Expand $S$ to a pants decomposition $\bar{S}=\\{\alpha_{i}\\}$,
then the Fenchel-Nielsen coordinates are given by the associated length and
twist parameters $\\{l_{i},\tau_{i}\\}$. For a small neighborhood $U$ of $X$
that is stratifically closed, let $(U,\\{k_{i}e^{i\tau_{i}}\\})$ be a chart,
with $k_{i}e^{i\tau_{i}}=l_{i}^{\chi}e^{i\tau_{i}}$ being the transition maps
on the overlap, where $\chi=\frac{1}{2}$ if $\alpha_{i}\in S$ and 1 otherwise.
We call such chart a nodal chart at $X$. Different extensions are
$C^{\infty}$-compatible by analytical compatibility of Fenchel-Nielsen
coordinates given by different pants decompositions.

Let $S=\\{\beta_{1},\cdots,\beta_{s}\\}$ be a set of disjoint simple closed
geodesics (not necessarily shortest) on $X$. As all $l_{\beta_{i}}\to 0$, by
definition $X\to X_{0}\in\partial\mathcal{M}$ ending in the respective
stratum. The limit nodal surface $X_{0}$ has a different topology and may not
be connected away from the nodes.

The tangent space of the compactification at $X_{0}\in\partial\mathcal{M}$ can
be decomposed as $T(X_{0})=T^{\textit{Str}}(X_{0})\oplus
T^{\textit{Nod}}(X_{0})$, where $T^{\textit{Str}}(X_{0})$ is given by the
length and twist parameters for $\bar{S}\setminus S$ and
$T^{\textit{Nod}}(X_{0})$ is given by those for $S$.

Under a nodal chart at some $X\in\partial\mathcal{M}$ with pinched curve set
$S$ as above, the geodesic-length functions $l_{\beta_{i}}=k_{\beta_{i}}^{2}$
are $C^{\infty}$.

To extend $\operatorname{sys_{T}}$ to $\partial\mathcal{M}$, note that for
fixed $T>0$,

$e^{-\frac{1}{T}l_{\beta_{i}}}\to 1\text{ as }l_{\beta_{i}}\to 0.$

For any $\gamma$ that crosses some $\beta_{i}$, since
$\sinh\frac{l_{\beta_{i}}}{2}\sinh\frac{l_{\gamma}}{2}>2\operatorname{arcsinh}1$,

$l_{\gamma}\to\infty\text{ and }e^{-\frac{1}{T}l_{\gamma}}\to 0\text{ as
}l_{\beta_{i}}\to 0.$

######  Definition 6.1.

For $X\in\partial\mathcal{M}$, let

$\displaystyle\operatorname{sys_{T}}(X)=$
$\displaystyle-T\log\left(s+\sum_{\gamma\text{ s.c.g. on
}X}e^{-\frac{1}{T}l_{\gamma}(X)}\right)$ $\displaystyle=$
$\displaystyle-T\log\left(s+\sum_{i}e^{-\frac{1}{T}\operatorname{sys_{T}}(X_{i})}\right),$

where $s=\\#S$.

It is known from previous discussion that $\operatorname{sys_{T}}$ is at least
$C^{2}$ on each stratum. Note that an extension of the Weil-Petersson metric
from $\mathcal{M}$ to the compactification $\overline{\mathcal{M}}$, that
equals the Weil-Petersson metric when restricted on each stratum, does not
exist. But given continuity of
$\operatorname{sys_{T}}|_{\overline{\mathcal{M}}}$, following the stratum-wise
gradient flow, we know that it attains its global minimum on maximally pinched
surfaces, and therefore is positively bounded on $\partial M$. We write this
as a lemma:

######  Lemma 6.2.

The function $\operatorname{sys_{T}}$ is bounded on $\overline{\mathcal{M}}$
by positive numbers.

######  Lemma 6.3.

For $T$ sufficiently small,
$\operatorname{sys_{T}}:\overline{\mathcal{M}}\to\mathbb{R}$ is $C^{2}$, under
the differential structure given by nodal charts.

###### Proof.

It is already shown that $\operatorname{sys_{T}}$ is $C^{2}$ on the interior
and continuous globally by its definition. Let $X\to
X_{0}\in\partial\mathcal{M}$ with pinched curve set
$S=\\{\beta_{1},\cdots,\beta_{s}\\}$. Recall that the first derivative is
given by

$\operatorname{sys_{T}}^{\prime}=\frac{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}l_{\gamma}^{\prime}}{\sum_{\gamma}e^{-\frac{1}{T}l_{\gamma}}},$

and the second derivative

$\displaystyle\operatorname{sys_{T}}^{\prime\prime}=\frac{1}{T}\left(\frac{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}l_{\gamma}^{\prime}}{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}}\right)^{2}-\frac{1}{T}\frac{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}(l_{\gamma}^{\prime})^{2}}{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}}+\frac{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}l_{\gamma}^{\prime\prime}}{\sum_{\gamma}e^{-{\frac{1}{T}}l_{\gamma}}}.$

Thus it suffices to show that $e^{-\frac{1}{T}}l^{\prime}$ and
$e^{-\frac{1}{T}}l^{\prime\prime}$ converge when the base point approaches the
boundary. We show that based on the following three types of geodesics:

Figure 10. Three types of $\gamma$

Type 1: $\gamma=\beta_{i}$. It is clear by definition that
$l=k_{\beta_{i}}^{2}$ is $C^{\infty}$.

Type 2: $\gamma\cap\beta_{i}=\emptyset$. We have
$l_{\gamma}=2\operatorname{arccosh}(\frac{\text{tr}(\Pi A_{i})}{2})$ by a
tracking algorithm by unfolding the surface, where each $A_{i}$ is a rotation
or translation matrix smooth in Fenchel-Nielsen coordinates and not involving
$\tau_{\beta_{i}}$’s.

Type 3: $i(\gamma,\cup\beta_{i})\neq 0$. We have $l_{\gamma}\geq-2\log
l_{\beta_{i}}$ for $\beta_{i}\cap\gamma\neq\emptyset$. Let $\gamma_{i}$’s be a
few geodesics disjoint from $\cup\beta_{i}$, such that $\\{\lambda_{i}=\nabla
l_{\beta_{i}}^{\frac{1}{2}},J\lambda_{i},\nabla l_{i}\\}$ is a local frame.

For the first directional derivatives, note that

$\displaystyle e^{-\frac{1}{T}l_{\gamma}}\left|\left\langle\nabla
l_{\gamma},\lambda_{i}\right\rangle\right|,e^{-\frac{1}{T}l_{\gamma}}|\left\langle\nabla
l_{\gamma},J\lambda_{i}\right\rangle|$ $\displaystyle\leq
e^{-\frac{1}{T}l}\|\nabla l_{\gamma}\|\cdot\|\lambda_{i}\|$ $\displaystyle\leq
e^{-\frac{1}{T}l_{\gamma}}\left(c\left(l_{\gamma}+l_{\gamma}^{2}e^{\frac{l_{\gamma}}{2}}\right)\right)^{\frac{1}{2}}\cdot
c\to 0,$

and

$\displaystyle e^{-\frac{1}{T}l_{\gamma}}\left|\left\langle\nabla
l_{\gamma},\nabla l_{i}\right\rangle\right|$ $\displaystyle\leq
e^{-\frac{1}{T}l}\|\nabla l_{\gamma}\|\cdot\|\nabla l_{i}\|$
$\displaystyle\leq
e^{-\frac{1}{T}l_{\gamma}}\left(c\left(l_{\gamma}+l_{\gamma}^{2}e^{\frac{l_{\gamma}}{2}}\right)\right)^{\frac{1}{2}}\cdot\left(c\left(l_{i}+l_{i}^{2}e^{\frac{l_{i}}{2}}\right)\right)^{\frac{1}{2}}\to
0.$

For the second directional derivatives, note that

$\displaystyle\left\lVert e^{-\frac{1}{T}l_{\gamma}}XYl_{\gamma}\right\rVert$
$\displaystyle=\left\lVert
e^{-\frac{1}{T}l_{\gamma}}\left(H(l_{\gamma})(X,Y)+\left\langle D_{X}Y,\nabla
l_{\gamma}\right\rangle\right)\right\rVert$ $\displaystyle\leq
e^{-\frac{1}{T}l_{\gamma}}\left(\left\lVert
H(l_{\gamma})(X,Y)\right\rVert+\left\lVert
D_{X}Y\right\rVert\cdot\left\lVert\nabla l_{\gamma}\right\rVert\right)$
$\displaystyle\leq
e^{-\frac{1}{T}l_{\gamma}}\left(\left(4c(1+l_{\gamma}e^{\frac{l}{2}}\right)+O\left(l_{\gamma}^{3}\right)\right)\left\lVert
X\right\rVert\cdot\left\lVert
Y\right\rVert+\frac{c}{l_{\beta_{i}}^{\frac{1}{2}}}\cdot\left\lVert\nabla
l_{\gamma}\right\rVert)\to 0,$

where $X,Y$ are two vector fields near $X_{0}$.

Therefore, $\sum_{\gamma\in\textit{Dt}(\gamma)}e^{\frac{1}{T}l_{\gamma}}$ is
$C^{2}$, where $\textit{Dt}(\gamma)$ is the set of $\gamma^{\prime}$’s that
are Dehn twist equivalent to $\gamma$ along $\cup\beta_{i}$. ∎

It is relatively clear about
$\operatorname{Crit}(\operatorname{sys_{T}}|_{\overline{\mathcal{M}}})\cap\mathcal{M}$.
For critical points in the boundary $\partial\mathcal{M}$, they correspond
exactly to the critical points for
$\operatorname{sys_{T}}|_{\mathcal{M}(\mathcal{S})}$. The
$T^{\textit{Str}}$-directional derivatives are 0 since
$\nabla\operatorname{sys_{T}}|_{\mathcal{S}}$ is parallel to
$\nabla\operatorname{sys_{T}}|_{\mathcal{M}(\mathcal{S})}$ under the
identification $\mathcal{S}\leftrightarrow\mathcal{M}(\mathcal{S})$ and so are
$T^{\textit{Nod}}$-directional derivatives following Remark 5.31.

On a nodal chart $(U,\\{k_{i}e^{i\tau_{i}}\\})$, the second derivative matrix
is

$\displaystyle\begin{pmatrix}2I&0\\\ 0&\ast\end{pmatrix},$

where $\ast$ is the second derivative matrix of $\operatorname{sys_{T}}$
restricted on $T^{\textit{Str}}$ that is nondegenerate. This shows that
critical points on $\partial\mathcal{M}$ are nondegenerate.

Based on the discussion above, we can give a description of critical piints in
the boundary $\partial\mgn$. Let $\mathcal{S}\subset\partial\mgn$ be a stratum
in the boundary, and write $\mathcal{S}=\Pi\mathcal{S}_{i}$ as a decomposition
by connected components of the base surface away from the nodes, with each
$\mathcal{S}_{i}$ isomorphic to some moduli space $\mathcal{M}_{i}$. Any
critical point $X\in\mathcal{S}$ is a nodal surface and has the corresponding
decomposition $X=\cup X_{i}$ plus the nodes, such that each $X_{i}$ is
critical in $\mathcal{M}_{i}$, and

$\operatorname{ind}(X)=\sum\operatorname{ind}(X_{i}),$

where $\operatorname{ind}(X_{i})$ is the index of $X_{i}$ for
$\operatorname{sys_{T}}|_{\mathcal{M}_{i}}$. This way we can decompose a
critical point in the boundary to smaller surfaces that are critical in their
respective moduli spaces. We can also construct a critical point by connecting
smaller critical points along nodes.

Figure 11. A critical point in $\partial\mathcal{M}_{1,4}$ that is the union
of smaller critical points in $\mathcal{M}_{1,1},\mathcal{M}_{0,3}$ and
$\mathcal{M}_{0,4}$ respectively along nodes

The surface $X$ in Figure 11 lives in
$\mathcal{S}=\mathcal{S}_{1}\times\mathcal{S}_{2}\times\mathcal{S}_{3}\subset\partial\mathcal{M}_{1,4}$,
where $\mathcal{S}_{1}\cong\mathcal{M}_{1,1}$,
$\mathcal{S}_{2}\cong\mathcal{M}_{0,3}$ and
$\mathcal{S}_{3}\cong\mathcal{M}_{0,4}$. Each subsurface off the nodes
$X_{i}\in\mathcal{S}_{i}$ is a critical point for
$\operatorname{sys}|_{\mathcal{S}_{i}}$ with $S(X_{i})$ marked in colors. The
index of $X$ is

$\operatorname{ind}(X)=\sum\operatorname{ind}(X_{i})=1+0+1=2.$

######  Theorem 6.4.

$\operatorname{sys_{T}}:\overline{\mathcal{M}}\to\mathbb{R}$ is $C^{2}$-Morse
with the given differential structure.

Part (3) of the main theorem follows the defined extension by a direct
calculation. Part (4) holds true for the extended $\operatorname{sys_{T}}$ for
similar reasons as in Subsections 5.2-5.5.

##  7\. Weil-Petersson Gradient Flow

As it is $C^{0}$, the systole function does not produce the Weil-Petersson
gradient vector field. Although one can consider the gradient ‘direction’ at
each point instead, it turns out to be a discontinuous distribution in the
tangent space. The $\operatorname{sys_{T}}$ functions are $C^{2}$, so a
gradient vector is defined wherever the Weil-Petersson metric is defined.

Note that the Weil-Petersson metric is well defined stratum-wise, but not on
the compactification $\mgnb$ as a whole. However, any flow line in a stratum
either stays in the same, or hits a critical point in its boundary in the
limit seb with infinite amount of time. Therefore, the Weil-Petersson gradient
flow of $\operatorname{sys_{T}}$ is perfectly well defined, and we can observe
the structure of the base space through the gradient flow.

######  Theorem 7.1.

The Weil-Petersson gradient flow of $\operatorname{sys}_{T}$ on the 6-sheeted
covering space of $\overline{\mathcal{M}}_{1,1}$ when $T$ is sufficiently
small:

Figure 12. Weil-Petersson gradient flow of $\operatorname{sys_{T}}$ on
$\overline{\mathcal{M}}_{1,1}$

The 3-punctured sphere is a 6-sheeted covering space of $\mathcal{M}_{1,1}$
that resolves the singularities. There are two red points lying at the center
of the front and the back respectively, and each is a triple point in the
moduli space where $\operatorname{sys_{T}}$ attains its maximum. Three blue
points are on the middle of the edges, each of which is a double point. See
[13] for more on these surfaces. The punctures correspond to 3-punctured
spheres that form the Deligne-Mumford boundary.

## Commonly Used Notations

$J$ |  multiindex, Weil-Petersson almost complex structure
---|---
$K$ |  multiindex
$F_{J}$ |  fan associated to $J$
$D$ |  constant, Weil-Petersson connection
$\gamma,\gamma_{i}$ |  simple closed geodesic
$\beta_{i}$ |  pinched curve
$l_{\gamma},l_{i}$ |  length function associated to $\gamma$ or $\gamma_{i}$
$m,M$ |  minimum, maximum of $\\{\|\nabla l_{i}\|,\|\nabla^{2}l_{i}(\tau,\cdot)\\}_{i;\tau}$
$\operatorname{sys}$ |  systole function
$\operatorname{sec.sys}$ |  second systole function
$\operatorname{sys_{T}}$ |  $:=-T\log\sum_{\gamma\text{ simple closed geodesic on }X}e^{-\frac{1}{T}l_{\gamma}(X)}$
$S$ |  pinched curve set
$S(X)$ |  set of shortest geodesics on a surface
$s_{X}(L)$ |  number of simple closed geodesics of length $\leq L$ on $X$
$r$ |  number of shortest geodesics
$I$ |  index set for $S(X)$
$I_{e}$ |  subset of $I$ indexing the maximal eutactic subset
$I_{b}$ |  complement of $I_{e}$ in $I$ when $X$ is semi-eutactic
$\mathbf{C}$ |  intersection of cones
$\operatorname{exp}_{p}$ |  exponential map at $p$
$i$ |  an automorphism of $T_{p}\mathcal{T}$
$\operatorname{ind}$ |  Morse index of a critical point
$\mathcal{T},\mathcal{M}$ |  Teichmüller space, moduli space
$\overline{\mathcal{M}},\partial\mathcal{M},\mathcal{S}$ |  Deligne-Mumford compactification, D-M boundary, stratum
$T^{\operatorname{sys}},T_{p}^{\operatorname{sys}}\mathcal{T}$ |  major subspace
$T^{\operatorname{sys}\perp},T_{p}^{\operatorname{sys}\perp}\mathcal{T}$ |  minor subspace
$T^{e},T^{e\perp\operatorname{sys}}$ |  subspace spanned by $\\{\nabla l_{i}\\}_{I_{e}}$, orthogonal complement of $T^{e}$ in $T^{\operatorname{sys}}$
$\tilde{\Omega}_{T},\Omega_{T}$ |  see Definition 4.7
$\tilde{\Phi}_{T},\Phi_{T}$ |  see Subsection 5.1
$\tilde{\Psi}_{T},\Psi_{T}$ |  see Definition 5.5
$\tilde{H}_{T},H_{T}$ |  see Subsection 5.3

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