# Short Simple Geodesic Loops on a 2-Sphere

Isabel Beach

###### Abstract

The classic Lusternik–Schnirelmann theorem states that there are three
distinct simple periodic geodesics on any Riemannian 2-sphere $M$. It has been
proven by Y. Liokumovich, A. Nabutovsky and R. Rotman that the shortest three
such curves have lengths bounded in terms of the diameter $d$ of $M$. We show
that at any point $p$ on $M$ there exist at least two distinct simple geodesic
loops (geodesic segments that start and end at $p$) whose lengths are
respectively bounded by $8d$ and $14d$.

##  1 Introduction

A geodesic loop based at a point $p$ in a Riemannian manifold $M$ is a
geodesic $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=p$. At each point $p$
on a Riemannian 2-sphere $M$, there are infinitely many distinct geodesic
loops based at $p$, as shown by J. P. Serre in [13]. Length bounds for
geodesic loops have been studied by many authors, including S. Sabourau in
[12] and F. Balacheff, H. Parlier and Sabourau in [1]. In [8, 9, 11], A.
Nabutovsky and R. Rotman produced length bounds for these geodesic loops in
terms of the geometric properties of $M$, culminating in the result that on a
Riemannian manifold $M$ with diameter $d$ diffeomorphic to $S^{2}$, there are
$k$ distinct geodesic loops based at any $p\in M$ of length at most $20kd$ for
every positive integer $k$ [9]. Building on this work, H. Y. Cheng improved
the bound to $6kd$ in general and to $5kd$ on a generic set of metrics on
$S^{2}$ when $k$ is odd [2]. Rotman also proved in [11] that the shortest
geodesic loop at any point on a closed manifold of dimension $n$ has length at
most $2nd$. Thus the shortest two non-trivial geodesic loops at any point on a
Riemannian 2-sphere are known to have lengths at most $4d$ and $12d$.

Similarly, a closed (or periodic) geodesic is a geodesic curve
$\gamma:\mathbb{R}\to M$ that is also periodic, and hence has an image that
can be parameterized by a geodesic loop. In this context, Rotman and
Nabutovsky proved a quantitative version of the Lusternik–Schnirelmann
theorem, showing that on a Riemannian 2-sphere of diameter $d$ there are
always two simple closed geodesics of lengths at most $5d$ and $10d$ [10].
With Y. Liokumovich, they upgraded this result to prove the existence of a
third simple closed geodesic of length at most $20d$ [5]. Note that because
these results deal with closed geodesics, they do not produce curves with a
specified base point.

In this paper, we are interested in the existence of simple geodesic loops of
bounded length. We will prove the following theorem.

######  Theorem 1.

Let $p$ be a point in an analytic Riemannian 2-sphere $M$ of diameter $d$.
Then there at least two simple geodesic loops based at $p$ of length at most
$8d$ and $14d$.

This differs from previous results regarding geodesic loops because it
concerns simple curves. Consequently, the two loops obtained in the proof of
our theorem will have distinct images. This is in contrast to the work of
Cheng, Nabutovsky and Rotman, which does not preclude the possibility that the
$k$ short geodesic loops are given by iterating the same closed geodesic $k$
times. This result also differs from the quantitative Lusternik–Schnirelmann
theorem in that it produces simple loops based at any point in $M$.

Our proof is built on the Lusternik–Schnirelmann proof of the existence of
three simple geodesics on the 2-sphere while also applying the quantitative
techniques used by Nabutovsky and Rotman. We follow the versions of the
Lusternik–Schnirelmann proof given by J. Hass and P. Scott in [3] and W.
Klingenberg in [4]. In order to produce three simple closed geodesics, the
standard technique is to use representatives of three non-trivial
$\mathbb{Z}_{2}$ homology classes in the space of unparameterized, unoriented
simple closed curves in $M$. In the cited proofs, one then iteratively applies
a shortening process to the curves in the image of each cycle. A subsequence
of curves from each cycle will then eventually converge to a non-trivial
closed geodesic. Using the fact that these three homology classes are related
by the cup product, it can be shown that either the three limiting geodesics
obtained in this manner are distinct, or that there are infinitely many such
geodesics.

Our paper is organized as follows. In Section 2.1, for any fixed $p\in M$ we
develop a curve shortening procedure that depends on $p$ and is based on the
disk flow of J. Hass and P. Scott [3]. We call our new process the modified
disk flow at $p$. Instead of unparameterized simple closed curves, we will
apply this process to what we call the $p$-admissible curves. Roughly
speaking, these are parameterized closed curves based at $p$ that can be
approximated by simple curves (see Definition 1). We will prove that the
modified disk flow preserves the set of $p$-admissible curves (Lemma 3) and
does not increase the lengths of these curves (Lemma 4). Importantly, we will
show that the image of a $p$-admissible curve under this process converges (up
to a subsequence) to a union of simple geodesic loops based at $p$ (Lemma 5).
We then describe how to extend this procedure to families of $p$-admissible
curves (Lemma 9) so that we can apply it simultaneously to all curves in the
image of a cycle.

Next, in Section 2.2 we discuss how to construct a continuous map of $S^{2}$
into $M$ of non-zero degree that sends the meridian curves on $S^{2}$ to
curves on $M$ that start at $p$ and end at some other point $q$. We call this
map a short meridional slicing of $M$, and we will use these curves to form
two homology cycles in the space of $p$-admissible curves. We first explain
how to shorten a closed curve consisting of a pair of minimizing geodesics by
applying the modified disk flow and exploiting the structure of the cut locus
(Lemma 12). We then pick a point $q$ at locally maximal distance from $p$ and
divide $M$ into disks bounded by minimizing geodesics connecting $p$ to $q$
using Berger’s Lemma. Using our previous lemma, we contract the boundaries of
these disks and turn the resulting homotopies into a short meridional slicing
of $M$ (Lemma 15). Critically, the lengths of these curves will be bounded by
the diameter of $M$.

Finally, in Section 2.3, we prove the existence of a pair of short simple
geodesic loops at $p$. We first show how to convert our short meridional
slicing into a sweepout of $M$ by short $p$-admissible curves based at $p$.
After some minor modifications, we will then have two non-trivial cycles: a
one-dimensional cycle formed by the slicing loops, and a two-dimensional cycle
formed by pairs of slicing loops. We conclude by applying the modified disk
flow to the images of the cycles (Lemma 16). Because these cycles are non-
trivial, some subsequence of curves in each image converges to a union of
short, non-trivial simple geodesic loops based at $p$. Following the
Lusternik–Schnirelmann proof, we then show that there are at least two
distinct such loops.

##  2 Results

###  2.1 The Modified Disk Flow

Our first step will be to create our curve shortening procedure. The original
disk flow developed by Hass and Scott in [3] proceeds on a compact manifold
$M$ as follows. Supposing we wish to shorten a closed curve $\gamma$, the disk
flow produces a family of curves $\gamma_{t}$ for $t\geq 0$ with
$\gamma_{0}=\gamma$. First, cover $M$ by a finite collection of totally normal
metric balls $\\{B_{i}\\}_{i=1}^{n}$ (i.e., metric balls with the property
that every point in their closure is connected by a unique minimizing geodesic
that lies in the ball itself). In addition, the collection of metric balls
with the same centres but half the radii should also cover $M$. We will
consider the balls $B_{i}$ consecutively, starting at $B_{1}$. If $\gamma\cap
B_{1}$ is empty, we do nothing. If instead $\gamma$ lies entirely in $B_{1}$,
then it can be contracted to a point curve in $B_{1}$. Otherwise, consider
each arc of $\gamma\cap B_{1}$ that connects two points on $\partial B_{1}$.
Because $B_{1}$ is totally normal, there is a unique geodesic connecting the
endpoints of each such arc. Moreover, this geodesic, excluding its endpoints,
lies in $B_{1}$. We redefine $\gamma$ to be the curve obtained by replacing
the arcs of $\gamma\cap B_{1}$ with these geodesics. We then repeat this
process in each consecutive ball. The resulting curve is defined to be
$\gamma_{1}$. We define $\gamma_{t}$ for $t\in(0,1)$ to be a suitable homotopy
between $\gamma$ and $\gamma_{1}$ – namely, a homotopy that does not increase
the number of self-intersections. We then iterate this entire procedure to
define $\gamma_{t}$ for $t\in[n,n+1]$, $n\in\mathbb{N}$, by repeating the
above with $\gamma_{n}$ in place of $\gamma$. Finally, there is a subsequence
of the $\gamma_{t}$ whose limit is a (possibly trivial) closed geodesic.
Moreover, this geodesic is necessarily shorter than $\gamma$.

When straightening in $B_{i}$, if the original arcs of $\gamma\cap B_{i}$ do
not intersect then neither will the minimizing geodesic segments they are
replaced with. This is because two minimizing geodesic segments cannot
mutually intersect more than once in their interiors, and they cannot
intersect non-transversely. Moreover, since our interpolating homotopy does
not increase the number of self-intersections, the disk flow preserves simple
curves. Thus it is possible to obtain a simple closed geodesic by starting
with a simple curve.

This flow can also be extended to multi-parameter families of curves. However,
discontinuities in the disk flow can occur when one curve being shortened is
tangent to the boundary of one of the balls in the cover and a nearby curve
lies entirely within said ball. While this situation can be avoided for a
single curve by slightly altering the cover, once we pass to a family of
curves we cannot necessarily arrange the disks to avoid tangencies in every
curve. Instead, each discontinuity is fixed by “filling in the gap” with an
interpolating homotopy.

Because we are interested in finding loops based at $p$, we want to modify the
disk flow so that it fixes the base point of the curve we are shortening.
Although it would be ideal to deal only with simple curves based at $p$, we
will need to examine what we call $p$-admissible curves.

######  Definition 1 ($p$-Admissible Curves).

Let $p\in M$. We call a curve $\gamma:[0,1]\to M$ $p$-admissible if the
following conditions hold.

  1. 1.

$\gamma(0)=\gamma(1)=p$

  2. 2.

$\gamma$ has no self-intersections except perhaps at the point $p$.

  3. 3.

There exists a homotopy of arbitrarily small width between $\gamma$ and a
simple closed curve.

In particular, such curves have no transverse self-intersections.

We now describe how we will alter the disk flow to create what we call the
modified disk flow at $p$. Note that we will only define this flow on
$p$-admissible curves. We cover $M$ by balls $B_{i}$ as before. However, we
additionally require that the first ball $B_{1}$ in our cover is centred at
$p$, and that no other ball contains $p$. We will only make modifications to
the flow in $B_{1}$. In order to fix $p=\gamma(0)$, we would like to replace
the arc of $\gamma$ that contains $t=0$ with the two geodesics that connect
$p$ to the arc’s endpoints (we call such geodesics “rays” because they emanate
from the centre of the metric ball). However, unlike in the standard disk
flow, this can create new transverse self-intersections since it is possible
that another geodesic segment crosses both geodesic rays. Therefore we make
the following compromise. First, the arc in $B_{1}$ containing $t=0$ is always
replaced by a pair of geodesic rays. Secondly, if replacing an arc in $B_{1}$
by a minimizing geodesic would create transverse self-intersections with the
pair of rays incident to $\gamma(0)$, we instead replace it with the pair of
geodesic rays that connect its endpoints to $p$. All other arcs are replaced
by genuine minimizing geodesics as in the original disk flow (see Figure 1).

$p$$p$ Figure 1: An example of the modified disk flow in $B_{1}$. The dashed
arc contains the point corresponding to $t=0$.

The above determines $\gamma_{1}$, which we call the image of $\gamma$ under
the modified disk flow. As before, we define $\gamma_{t}$ for $t\in(0,1)$ to
be a suitable homotopy between $\gamma=\gamma_{0}$ and $\gamma_{1}$. The
homotopy $\gamma_{t}$ is in fact the same as in the original disk flow,
despite the fact that we allow geodesic rays. The original description of the
homotopy can be found in Lemma 1.6 of [3]. For completeness, we briefly
reprove it here for $p$-admissible curves.

######  Lemma 1.

A $p$-admissible curve can be homotoped to its image under the modified disk
flow through $p$-admissible curves.

###### Proof.

For brevity, let the image of $\gamma$ under the modified disk flow at $p$ be
denoted by $\eta$ (i.e, $\eta=\gamma_{1}$). It is enough to define a homotopy
in $B_{i}$ between $\gamma\cap B_{i}$ and $\eta\cap B_{i}$ for each $i$.
Enumerate the arcs of $\gamma\cap B_{i}$ and call them $\gamma_{i}^{j}$, and
let the corresponding arcs of $\eta\cap B_{i}$ be called $\eta_{i}^{j}$. To
ensure our homotopy is through $p$-admissible curves, we want to homotope
$\gamma_{i}^{j}$ to $\eta_{i}^{j}$ through arcs that do not intersect each
other except perhaps non-transversely at $p$.

If the region bounded by $\gamma_{i}^{j}*-\eta_{i}^{j}$ contained no other
arcs of $\gamma$, we could homotope $\gamma_{i}^{j}$ to $\eta_{i}^{j}$ without
creating intersections. By $p$-admissibility, if any other $\gamma_{i}^{k}$
crosses $\eta_{i}^{j}$, $k\not=j$, necessarily it crosses an even number of
times (excluding a possible tangency at $p$). Thus the region bounded by
$\gamma_{i}^{j}*-\eta_{i}^{j}$ is divided into regions bounded by an arc of
$\eta_{i}^{j}$ and an arc of $\gamma$. Call these regions digons. Of all the
digons cobounded by arcs of $\eta_{i}^{j}$, consider one that does not contain
any other digon cobounded by an arc of $\eta_{i}^{j}$. We can homotope the
other bounding arc of this digon to an arc of $\eta_{i}^{j}$ without
intersecting anything other arc of $\gamma$ (see Figure 2), thus removing this
digon. Then we slide the arc slightly so it lies outside the bounds of
$\gamma_{i}^{j}*-\eta_{i}^{j}$ and does not intersect $\eta_{i}^{j}$. Repeat
this process until all the digons cobounded by $\eta_{i}^{j}$ inside
$\gamma_{i}^{j}*-\eta_{i}^{j}$ have been removed. We can then homotope
$\gamma_{i}^{j}$ to $\eta_{i}^{j}$ without introducing intersection points.
Repeat this process for every $j$. ∎

$\eta_{i}^{j}$$\gamma$ Figure 2: Eliminating a digon in order to homotope an
arc to the corresponding geodesic segment. The dashed line is $\eta_{i}^{j}$
and the solid line is an arc of $\gamma$.

As before, we define $\gamma_{t}$ for $t\in[n,n+1]$, $n\in\mathbb{N}$ by
iterating the modified disk flow procedure described above.

Now that we have defined our procedure, we prove a lemma that clarifies some
of its behaviour.

######  Lemma 2.

Let $\gamma$ be a $p$-admissible curve in $M$. Suppose the arc $\gamma_{0}$ of
$\gamma\cap B_{1}$ containing $\gamma(0)$ intersects $\partial B_{1}$ in two
points. Suppose there is another arc of $\gamma\cap B_{1}$, say
$\gamma^{\prime}$. Let $B^{\prime}$ be the arc of $\partial B_{1}$ connecting
the endpoints of $\gamma^{\prime}$ that does not contain the endpoints of
$\gamma_{0}$. The arc $B^{\prime}$ subtends an angle $\theta$ in $\partial
B_{1}$. Under the modified disk flow, $\gamma^{\prime}$ will be replaced by a
minimizing geodesic if and only if $\theta\leq\pi$.

###### Proof.

Note that such an arc $B^{\prime}$ exists because $\gamma$ is $p$-admissible.

Denote the geodesic rays connecting $p$ to the endpoints of $\gamma_{0}$ by
$\alpha$ and $\beta$. We need to determine when the minimizing geodesic $\eta$
connecting the endpoints of $\gamma^{\prime}$ crosses $\alpha\cup\beta$.
Denote the geodesic rays connecting $p$ to the endpoints of $\gamma^{\prime}$
by $\alpha^{\prime}$ and $\beta^{\prime}$. If $\theta\leq\pi$, then
$\alpha^{\prime}\cup\beta^{\prime}\cup B^{\prime}$ bounds a convex region, and
hence the closure of this region contains $\eta$. On the other hand, this
region does not contain $\alpha\cup\beta$, so $\eta$ will not intersect them
(except perhaps non-transversely at $p$ if $\theta=\pi$). Thus
$\gamma^{\prime}$ will be replaced by $\eta$ under the modified disk flow.
Alternatively, if $\theta>\pi$, then the complement of the region bounded by
$\alpha\cup\beta$ is convex, and hence the complement contains both $\eta$ and
$\alpha\cup\beta$. Necessarily, $\eta$ must cross both $\alpha$ and $\beta$
transversely. Thus $\gamma^{\prime}$ will be replaced by two rays through $p$
under the modified disk flow. ∎

The rest of this section is dedicated to proving certain properties of the
modified disk flow. In order to successfully apply the Lusternik–Schnirelmann
proof, we require the following.

  1. 1.

The modified disk flow must preserve $p$-admissibility.

  2. 2.

Under repeated application of the modified disk flow, a curve must converge
(up to a subsequence) to either the point curve $p$, to a union of at least
two distinct simple geodesic loops at $p$, or to a single prime simple
geodesic loop at $p$.

  3. 3.

Under the modified disk flow, a curve’s length is strictly decreased unless it
is a union of geodesic loops at $p$.

  4. 4.

The modified disk flow can be extended to one- and two-parameter families of
$p$-admissible curves.

  5. 5.

A family of $p$-admissible curves must be homotopic via $p$-admissible curves
to its image under the modified disk flow.

We first prove that the modified disk flow preserves $p$-admissibility.

######  Lemma 3.

Let $\gamma$ be a $p$-admissible curve in $M$. Then the image of $\gamma$
under the modified disk flow is also $p$-admissible.

###### Proof.

Because we are replacing arcs with minimizing geodesics, the original disk
flow does not create transverse intersections. This is shown in Theorem 1.8 of
[3]. Therefore, because $p$-admissible curves have self-intersections only in
$B_{1}$ and the only differences between the original and modified disk flows
occur in $B_{1}$, we will only consider what happens in $B_{1}$. If $\gamma$
lies entirely in $\overline{B_{1}}$, then its image under the disk flow is the
point $p$. Otherwise, the arc $\gamma_{0}$ of $\gamma\cap B_{1}$ containing
$\gamma(0)$ intersects $\partial B_{1}$ in two distinct points. Denote the two
geodesic rays that connect these two points to $p$ by $\alpha$ and $\beta$.

Apply the modified disk flow to $\gamma$ in $B_{1}$. Because $\gamma$ is
$p$-admissible, if we obtain a transverse self-intersection after the modified
disk flow it must be due to a minimizing geodesic crossing a pair of rays. By
Lemma 2, every ray pair obtained by applying the modified disk flow bounds a
convex sector of $B_{1}$ that contains $\alpha$ and $\beta$. Therefore a
minimizing geodesic that crosses any ray pair transversely must also cross
$\alpha$ and $\beta$ transversely (see Figure 3). This contradicts the fact
that in such a case we would have replaced the original arc corresponding to
this minimizing geodesic with a pair of rays through $p$ instead. Thus there
can be no transverse self-intersections in $B_{1}$.

Suppose we obtain a non-transverse self-intersection. After applying the
modified disk flow, $\gamma\cap B_{1}$ consists of minimizing geodesic arcs
connecting either a pair of points of $\partial B_{1}$ or $p$ and a point of
$\partial B_{1}$. Therefore a non-transverse intersection occurs only when two
geodesic segments of $\gamma$ coincide, meet at a point on $\partial B_{1}$,
or meet at $p$. By definition, non-transverse intersections at $p$ are allowed
for a $p$-admissible curve. The other two cases occur when two arcs of the
original $\gamma\cap B_{1}$ share either one or both endpoints on $\partial
B_{1}$, which does not occur because $\gamma$ only has self-intersections at
$p$. Thus $\gamma$ remains $p$-admissible after applying the modified disk
flow. ∎

$p$$\alpha$$\beta$$\alpha^{\prime}$$\beta^{\prime}$$\eta$ Figure 3: A
minimizing geodesic $\eta$ that crosses the pair of rays $\alpha^{\prime}$ and
$\beta^{\prime}$ must also cross the inner pair of rays $\alpha$ and $\beta$.

The remaining properties of a single curve under the modified disk flow will
be proved directly in a manner similar to the proof for the original disk flow
(Theorem 1.8 of [3]). Next we prove that the modified disk flow does not
increase the lengths of curves.

$p$$\gamma_{0}$$\gamma^{\prime}$ Figure 4: Left: there exists a diameter that
lies outside the sector given by the endpoints of $\gamma_{0}$ and intersects
$\gamma^{\prime}$ at least twice. Right: the curve obtained by replacing a
section of $\gamma^{\prime}$ with a portion of a diameter, which is shorter
than $\gamma^{\prime}$.

######  Lemma 4.

Let $\gamma$ be a $p$-admissible curve in $M$. Then the image of $\gamma$
under the modified disk flow is not longer than $\gamma$.

###### Proof.

The standard disk flow does not increase lengths since arcs are always being
replaced by minimizing geodesics with the same endpoints. Thus the only way we
might make $\gamma$ longer is when we replace an arc in $B_{1}$ with two rays
through $p$ that do not form a minimizing geodesic. Any arc of $\gamma\cap
B_{1}$ that passes through $p$ is not lengthened by replacing it with the two
rays that connect its endpoints to $p$. Therefore we only need to consider an
arc $\gamma^{\prime}$ with endpoints on $\partial B_{1}$ that does not pass
through $p$.

Suppose that under the modified disk flow $\gamma^{\prime}$ gets replaced by
two rays $\alpha^{\prime},\beta^{\prime}$ that do not form a minimizing
geodesic. Let $\gamma_{0}$ be the arc of $\gamma\cap B_{1}$ containing
$\gamma(0)$. As in Lemma 3, $\alpha^{\prime}$ and $\beta^{\prime}$ bound a
convex sector of angle strictly less than $\pi$ that contains the endpoints of
$\gamma_{0}$. Therefore we can pick a minimizing geodesic $\eta$ that passes
through $p$ and connects two points on $\partial B_{1}$ that lie outside of
this sector. Necessarily, the endpoints of $\gamma^{\prime}$ and $\gamma_{0}$
lie on the same side of $\eta$. The arcs $\gamma_{0}$ and $\eta$ both pass
through $p$, while $\gamma^{\prime}$ does not pass through $p$ but also does
not cross $\gamma_{0}$. Therefore $\gamma^{\prime}$ must intersect $\eta$ at
least twice (see Figure 4, left). Therefore replacing an arc of
$\gamma^{\prime}$ with an arc of $\eta$ that passes through $p$ creates a
curve no longer than $\gamma^{\prime}$ that passes through $p$ (see Figure 4,
right). We have now reduced to our preliminary case, and the result follows. ∎

We now show that a subsequence of the curves obtained by applying the modified
disk flow will converge to either a point curve or some number of simple
geodesic loops based at $p$. Again, we follow the proof structure for the
corresponding theorem for the original disk flow.

######  Lemma 5.

Let $\gamma$ be a $p$-admissible loop and let $\gamma_{t}$, $t\in[0,\infty)$,
denote the family obtained by applying the modified disk flow. Then the
following hold.

  1. 1.

A subsequence of the curves $\gamma_{i}$, $i\in\mathbb{N}$, either converges
to the point curve $p$ or to a union of simple geodesic loops
$\gamma_{\infty}$ at $p$. In the latter case,
$L(\gamma_{\infty})=\lim\limits_{i\to\infty}L(\gamma_{i})\leq L(\gamma)$.

  2. 2.

$L(\gamma_{i+1})=L(\gamma_{i})$ for some $i\in\mathbb{N}$ if and only if
$\gamma_{i}$ is a point curve or a union of simple geodesic loops at $p$.

###### Proof.

Lemma 4 proves that the lengths of the curves $\gamma_{i}$ is non-increasing,
and hence uniformly bounded by $L(\gamma)$. Therefore we can apply the
Arzèla–Ascoli theorem to conclude the existence of a convergent subsequence
with uniform limit $\gamma_{\infty}$ satisfying
$L(\gamma_{\infty})\leq\liminf_{t\to\infty}L(\gamma_{t})$. If
$\liminf_{t\to\infty}L(\gamma_{t})$ is zero, then $\gamma_{\infty}$ is
necessarily a point curve and
$L(\gamma_{\infty})=\lim\limits_{i\to\infty}L(\gamma_{i})$ as claimed.
Otherwise, assume $0<\liminf_{t\to\infty}L(\gamma_{t})$. We will first show
that $\gamma_{\infty}$ is a piecewise geodesic curve with vertices only at $p$
(i.e., a union of geodesic loops at $p$). Suppose this is not the case. Then
there are two possibilities. The first option is that $\gamma_{\infty}$
contains a piecewise geodesic subarc with a vertex at some $x\in\partial
B_{i}$ (note that $x\not=p$). The point $x$ lies in the interior of some
$B_{j}$, $j\not=i$. Then $\gamma_{\infty}$ can be strictly shortened by
applying the modified disk flow again, since the non-geodesic arc in $B_{j}$
containing $x$ will be replaced by a minimizing geodesic. The other
possibility is that $\gamma_{\infty}$ has a non-geodesic subarc lying entirely
inside some $B_{i}$ that is not formed of two minimizing geodesic rays through
$p$. Once again, another application of the modified disk flow will produce a
strictly shorter curve.

In either case, our new curve is of length $L(\gamma_{\infty})-\epsilon$ for
some $\epsilon>0$. We will show that this contradicts the fact that
$L(\gamma_{\infty})\leq\liminf_{t\to\infty}L(\gamma_{t})$. Choose $k$ large
enough so that $\gamma_{k}$ is within distance $\epsilon/10$ of
$\gamma_{\infty}$ in the sup norm. Then an application of the modified disk
flow must also strictly shorten $\gamma_{k}$, say by some $\delta>0$, since it
is very close to $\gamma_{\infty}$. By definition we have
$L(\gamma_{k})=L(\gamma_{k+1})+\delta$. Conversely, we know that
$L(\gamma_{l})-L(\gamma_{\infty})<\delta$ for any sufficiently large $l$, and
in particular for $k$. In combination, we have that
$L(\gamma_{k+1})<L(\gamma_{\infty})$, contradicting the fact that
$L(\gamma_{\infty})\leq\liminf_{t\to\infty}L(\gamma_{t})$. Thus
$\gamma_{\infty}$ must be a union of geodesic loops at $p$.

Note that the loops in the limit are also simple. This is because $\gamma_{t}$
is $p$-admissible for all $t<\infty$, and hence has only non-transverse self-
intersections at $p$. Therefore the same is true of $\gamma_{\infty}$. Because
geodesic segments never have non-transverse self-intersections, each loop of
$\gamma_{\infty}$ is simple. Of course, $\gamma_{\infty}$ may have self-
intersections at $p$.

We now address the second claim. If $L(\gamma_{i})=0$, then $\gamma_{i}$ is a
point curve and hence $\gamma_{j}$ is a point curve for all $j>i$. If
$L(\gamma_{i})>0$ and $\gamma_{i}$ is not of the desired form, then, as in the
proof of the first claim, an application of the modified disk flow will
produce a strictly shorter curve. However, this contradicts the fact that
$L(\gamma_{i})=L(\gamma_{i+1})$. ∎

From now on, we will let $\gamma_{\infty}$ denote the limit of a choice of
such a subsequence $\gamma_{i}$ from the statement of Lemma 5. Our ultimate
goal is to find two geometrically distinct geodesic loops based at $p$. Thus
we want to prove that if $\gamma_{\infty}$ is non-trivial, it is not given by
iterating a single loop multiple times. This is shown in the following two
lemmas.

######  Lemma 6.

Let $\gamma$ be a $p$-admissible loop. If $\gamma_{\infty}$ is given by
iterating only the (simple) geodesic loops $\eta$ and $-\eta$, then in fact
either $\gamma_{\infty}=\eta^{a}$ or $\gamma_{\infty}=-\eta^{a}$ for some
$a\geq 1$.

###### Proof.

Without loss of generality, orient $\eta$ so that the first loop of
$\gamma_{\infty}$ is $\eta$. We will show that no arc of $\gamma_{\infty}$ is
given by $\eta*-\eta$, and hence that $\gamma_{\infty}=\eta^{a}$ for some
$a\geq 1$. Suppose instead that some arc converged to $\eta*-\eta$. Consider
an arc of the form $\eta\mid_{[0,s]}*-\eta\mid_{[0,s]}$ for some small $s$. No
such arc is geodesic and no such arc is arbitrarily well-approximated by a
geodesic. Therefore the arcs of $\gamma_{i}$ that converge to
$\eta\mid_{[0,s]}*-\eta\mid_{[0,s]}$ (or at least the portion lying in
$B_{1}\setminus\cup_{i>1}B_{i}$) must consist of pairs of geodesic rays
through $p$. However, the fact that they are rays at all means that they must
bound a convex sector that encloses the arc of $\gamma_{i}\cap B_{1}$ passing
through $\gamma_{i}(0)$. Let this sector have angle $\theta_{i}$. Necessarily,
$\theta_{i}$ must tend to 0, which we will show leads to a contradiction. We
will consider two cases: when the final loop of $\gamma_{\infty}$ is $\eta$,
and when it is $-\eta$.

If the final loop of $\gamma_{\infty}$ is $\eta$, then the angle at
$\gamma_{\infty}(0)$ is greater than zero since it equals the angle of $\eta$
at $p$. Therefore for sufficiently large $i$ the angle at $\gamma_{i}(0)$ is
bounded from below by some $c>0$ (see Figure 5, left). Then $\theta_{i}$ must
be at least as large as $c$ because the corresponding sector contains the arc
of $\gamma_{\infty}$ through $p$. In particular, the $\theta_{i}$ cannot tend
to zero, which is a contradiction.

On the other hand, if the final loop of $\gamma_{\infty}$ is $-\eta$, then the
angle at $\gamma_{\infty}(0)$ is zero. In this case, consider an arc of
$\eta*-\eta$ in $B_{1}$ of the form $\eta\mid_{[r,1]}*-\eta\mid_{[r,1]}$ for
some small $r>0$. As before, this arc is not geodesic and so the arcs of
$\gamma_{i}$ that converge to it must also consist of pairs of geodesic rays
through $p$ whose angles at $p$ tend to 0 (see Figure 5, right). Therefore
these rays must also bound an acute sector containing the arc of
$\gamma_{\infty}$ through $\gamma_{\infty}(0)$. However, this sector is
disjoint from the previous one by $p$-admissibility, so they cannot both
contain the arc of $\gamma_{\infty}$ through $\gamma_{\infty}(0)$. Thus we
once again obtain a contradiction. ∎

$B_{1}$$c$$2\pi-\theta_{i}$$\gamma_{i}$$\eta$$B_{1}$$\theta_{i}$$\gamma_{i}$$\eta$
Figure 5: Left: the sector bounded by $\gamma_{i}$ when the final loop of
$\gamma_{\infty}$ is $\eta$. Right: the two sector bounded by $\gamma_{i}$
when the final loop of $\gamma_{\infty}$ is $-\eta$.

######  Lemma 7.

Let $\gamma$ be a $p$-admissible loop. If $\gamma_{\infty}=\eta^{a}$ for some
geodesic loop $\eta$, then $a=1$ (i.e., $\gamma_{\infty}$ is prime).

###### Proof.

Let $r$ be small enough that $B_{r}(p)$ is contained in
$B_{1}\setminus\cup_{i>1}B_{i}$ and $B_{r}(p)\cap\eta$ consists of exactly one
arc. This is possible because $\eta$ is simple by Lemma 5. Pick $\epsilon<r/2$
and let $i$ be large enough that $\gamma_{i}$ is within distance $\epsilon/2$
of $\gamma_{\infty}$ in the sup norm. Because $\gamma_{i}$ is $p$-admissible,
we can apply a homotopy of very small width to make $\gamma_{i}$ into a simple
curve that still lies in an $\epsilon$-neighbourhood of $\eta$. Consider the
arcs of $\gamma_{i}\cap B_{r}(p)$. We will call these arcs $\Gamma_{i}^{j}$,
$j\in\\{1,\cdots,k\\}$, ordered as follows. Letting $\partial B_{r}(p)$ be
positively oriented, choose some arc of $\gamma_{i}\cap B_{r}(p)$ that, with
$\partial B_{r}(p)$, cobounds a region of $B_{r}(p)$ whose interior contains
no other arc of $\gamma_{i}$. This follows from the fact that $\gamma_{i}$ is
simple. Call this arc $\Gamma_{i}^{1}$. Number the other arcs of
$\gamma_{i}\cap B_{r}(p)$ in the order in which they are encountered when
following $\partial B_{1}$ from the endpoint of $\Gamma_{1}^{i}$. This
ordering is well-defined because $\gamma_{i}$ has no transverse self-
intersections and self-intersects only at $p$. Moreover, let $x_{j}$ be the
initial point of the arc $\Gamma_{i}^{j}$ and let $y_{j}$ be the end point,
according to the orientation of $\gamma_{i}$. These points are necessarily
arranged along $\partial B_{1}$ in the cyclic order
$x_{1},x_{2},x_{3},\cdots,x_{k},y_{k},y_{k-1},\cdots,y_{2},y_{1}$.

$\eta_{x_{1},y_{m}}$$\gamma_{x_{1},y_{m}}$$x_{1}$$x_{l}$$x_{m}$$y_{1}$$y_{l}$$y_{m}$$\eta_{x_{l},y_{1}}$$\gamma_{x_{l},y_{1}}$$y_{1}$$y_{l}$$y_{m}$$x_{1}$$x_{l}$$x_{m}$
Figure 6: Separating the endpoints of arcs $\Gamma_{i}^{j}$ by
$\gamma_{x_{1},y_{m}}\cup\eta_{x_{1},y_{m}}$ (left) and
$\gamma_{x_{l},y_{1}}\cup\eta_{x_{l},y_{1}}$ (right).

Because $B_{r}(p)\cap\eta$ only contains the arc of $\eta$ passing through
$p$, for sufficiently small $\epsilon$ every $\Gamma_{i}^{j}$ represents an
arc of $\gamma_{i}$ that is converging to an arc of the form
$\eta\mid_{[1-s,1]}*\eta\mid_{[0,s]}$ for some $s$. Moreover, every arc of
$\gamma_{i}$ that converges to a copy of $\eta$ corresponds to exactly one arc
$\Gamma_{i}^{j}$. We will now prove that $k=1$, and hence that
$\gamma_{\infty}=\eta$. Following $\gamma_{i}$ forward from $y_{1}$, we next
encounter some $x_{l}$. Let the arc of $\gamma_{i}$ joining these two points
be denoted $\gamma_{y_{1},x_{l}}$, noting that this arc lies entirely outside
of $B_{1}$. Going backwards along $\gamma_{i}$, $x_{1}$ is followed by some
$y_{m}$. Let the arc of $\gamma_{i}$ joining these two points be denoted
$\gamma_{y_{m},x_{1}}$, which we note again lies outside $B_{1}$. If
$\rho_{x_{l},y_{1}}$ is the minimizing geodesic in $B_{1}$ connecting $x_{l}$
and $y_{1}$, then $x_{1}$ is separated from any $y_{i}$ with $i>1$ and any
$x_{j}$ with $j>l$ by the closed curve
$\gamma_{y_{1},x_{l}}*\rho_{x_{l},y_{1}}$ (see Figure 6, left). Similarly,
$y_{1}$ is separated from any $x_{i}$ with $i>1$ and any $y_{j}$ with $j>m$ by
the closed curve $\gamma_{y_{m},x_{1}}*\rho_{x_{1},y_{m}}$ (see Figure 6,
right). However, this means that the arcs $\gamma_{y_{1},x_{l}}$ and
$\gamma_{y_{m},x_{1}}$ must either cross, which they cannot by
$p$-admissibility, or they coincide (i.e., they are the same portion of
$\gamma$). Thus we must have $x_{l}=x_{1}$ and $y_{m}=y_{1}$, meaning that
$k=1$ and hence that $\gamma_{\infty}=\eta$. ∎

Consequently, we can find a subsequence of curves with decreasing lengths that
converges to either a point or a union of simple geodesic loops. If we obtain
loops, then either at least two of our loops have distinct images or we obtain
a single prime loop.

We now show how to apply the modified disk flow to one- or two-parameter
families of $p$-admissible curves. A $k$-parameter family of $p$-admissible
curves is a continuous map $f:X\to\Pi_{p}(M)$, where $X$ is a $k$-dimensional
manifold and $\Pi_{p}(M)$ is the space of parameterized $p$-admissible curves.
As in the original disk flow, if a curve in a family is ever tangent to one of
the circles $\partial B_{i}$, there may be discontinuities in the disk flow
process. This is because, under the (modified) disk flow, an arc with a
tangency will be replaced by a pair of geodesic segments, while a nearby arc
that lies entirely within the disk will be replaced by only one geodesic
segment. Thus, following [3], after we apply the modified disk flow we must
replace the tangent arc with a family of arcs that interpolates between the
pair of geodesics and the single geodesic (see Figure 7). In the case of a
curve with two tangent points, there may be a similar discontinuity when there
are curves with only one tangent point nearby. Note that, following Lemma 3.1
of [3], our covering can be modified slightly so that no curve in a one-
parameter family has more than one tangency point, and no curve in a two-
parameter family has more than two tangency points.

Figure 7: A family of curves in a disk before and after applying the
(modified) disk flow. One of the curves is tangent to the boundary. To fix the
resulting discontinuity, we must interpolate between the solid and dashed
curves bounding the grey triangle.

In order to fill in discontinuities, we need to interpolate between a
minimizing geodesic and a piecewise minimizing geodesic with the same
endpoints. This pair of curves collectively bounds a union of convex polygons
with geodesic edges. Thus we now show how to homotope from one edge of the
boundary of a convex polygon to the other edges while fixing endpoints.

######  Lemma 8.

Any edge of a convex geodesic polygon can be homotoped to the remaining edges
through a length-non-increasing homotopy with fixed endpoints. Moreover, this
homotopy depends continuously on the polygon.

###### Proof.

Let $\rho_{a,b}$ denote the unique minimizing geodesic connecting any two
points $a$ and $b$ in the polygon. If we want our homotopy to end at the
single edge $\rho_{x,y}$, let $u(t):[0,1]\to M$ be a parameterization of the
remaining edges of the polygon in question. Then we can simply take our
homotopy to be $t\mapsto u\mid_{[0,t]}*\rho_{u(t),u(1-t)}*u\mid_{[1-t,1]}$
(see Figure 8 for an example case). Note that these homotopies are
$p$-admissible, since these convex polygons necessarily do not contain any
other arcs of the curve in their interiors. Moreover, these homotopies are
independent of the orientation of the underlying curve, vary continuously with
respect to the original curves. They are also length non-increasing, since we
are replacing arcs of $u$ by minimizing geodesics. ∎

Figure 8: An example of an interpolating homotopy between edges of a geodesic
triangle.

We now define the modified disk flow on one- or two-parameter families of
$p$-admissible curves. Given a family $f:X\to\Pi_{p}(M)$, we will define a
homotopy $f_{t}:X\to\Pi_{p}(M)$, $t\in[0,1]$, between $f=f_{0}$ and some
$f_{1}$. We call $f_{1}$ the image of $f$ under an application of the modified
disk flow.

######  Lemma 9.

The modified disk flow can be extended to continuous one- and two-parameter
families of $p$-admissible curves.

###### Proof.

Consider the family $f:X\to\Pi_{p}(M)$, where $X$ is a one- or two-dimensional
manifold. As in the original disk flow, discontinuities occur when a curve
with one or more tangencies is arbitrarily close to a curve with fewer
tangencies. For ease, let $\rho_{a,b}$ denote the unique minimizing geodesic
connecting any two points $a,b$ in the ball $B_{i}$. There are only a finite
number of possible tangency point configurations (see Figure 9). If we want to
interpolate between a single tangency and no tangency, or between a single and
double tangency, we need to homotope between $\rho_{x,y}$ and
$\rho_{x,a}*\rho_{a,y}$ for some $x,y,a\in B_{i}$. Similarly, to interpolate
between a double tangency and no tangency we need to homotope between
$\rho_{x,y}$ and $\rho_{x,a}*\rho_{a,b}*\rho_{b,y}$ for some $x,y,a,b\in
B_{i}$.

Figure 9: Possible tangency configurations for one- and two-parameter families
of $p$-admissible curves. To fixed the resulting discontinuities, we must
homotope the solid arcs with tangencies to the corresponding dashed arcs.

In the first case, $\rho_{x,a}*\rho_{a,y}*-\rho_{x,y}$ always bounds a single
convex geodesic polygon. In the second case, it is possible that $\rho_{x,y}$
and $\rho_{a,b}$ intersect at some point $c$. When this happens, we can
instead homotope $\rho_{x,c}$ to $\rho_{x,a}*\rho_{a,c}$ and $\rho_{c,y}$ to
$\rho_{c,b}*\rho_{b,y}$. As before, $\rho_{x,a}*\rho_{a,c}*-\rho_{x,c}$ and
$\rho_{c,b}*\rho_{b,y}*-\rho_{y,c}$ each bound a convex geodesic polygon. If
there is no such intersection, then
$\rho_{x,a}*\rho_{a,b}*\rho_{b,y}*-\rho_{x,y}$ itself bounds a convex geodesic
polygon. Thus, in every possible case, our desired homotopy is obtained by
homotoping one edge of a convex geodesic polygon to the other edges. We can do
this by applying Lemma 8.

We define $f_{1}$, the image of $f$ under the modified disk flow, by applying
the modified disk flow to every curve $f(x),x\in X$, individually and then
making the following modifications. A point in the parameter space $X$
corresponding to a curve with a single tangency is replaced by an interval
parameterizing one of the above homotopies. A point corresponding to a curve
with a double tangency is replaced by a polygon parameterizing the following
family. One vertex of the polygon is mapped to the curve with two tangencies,
and the other vertices are mapped to adjacent curves with one tangency. The
edges parameterize the homotopy from Lemma 8 interpolating between pairs of
vertices. The family of curves parameterized by the polygon’s boundary can be
homotoped to a constant map using the homotopies in Lemma 8. Thus this map is
nullhomotopic in the space of $p$-admissible curves and therefore can be
extended to the interior of the polygon. For further details, see the proof of
Lemma 3.8 in [3]. Note that these homotopies do not introduce self-
intersections, since the regions we are “filling in” necessarily do not
contain any other arcs of the curve. Moreover, they are independent of the
parameterization and orientation of the underlying curve.

Now we will show that no discontinuities arise from our modifications to the
disk flow. As long as a continuous family of arcs (not tangent to $B_{1}$) are
all either replaced by rays or all replaced by geodesics, then the resultant
curves also form a continuous family because geodesics in $B_{1}$ vary
continuously as a function of their endpoints. Thus we only need to examine
the transition between these two cases. Recall that the curves in our family
are denoted by $f(x)$, $x\in X$. Let $f(x)_{0}$ denote the arc of $f(x)\cap
B_{1}$ through $p=(f(x))(0)$ and let $\alpha(x)$ and $\beta(x)$ denote the
geodesic rays connecting $p$ to the endpoints of $f(x)_{0}$. Let the other
arcs of $f(x)\cap B_{1}$ be denoted $f(x)_{i}$ for $i>0$. Then, as in Lemma 3,
the rays connecting $p$ to the endpoints of $f(x)_{i}$ bound a sector of
$B_{1}$ containing $\alpha(x)\cup\beta(x)$. This sector has angle
$\theta(x)_{i}$, and by Lemma 2 $f(x)_{i}$ is replaced by a minimizing
geodesic if and only if $\theta(x)_{i}\geq\pi$. Otherwise, it is replaced by
two rays. Note that the rays $\alpha(x)$ and $\beta(x)$ change continuously
with $x$ (assuming the absence of tangencies), as does each $\theta(x)_{i}$.
Therefore the family of arcs $f(x)_{i}$ can only transition between being
replaced by rays and replaced by geodesics when $\theta(x)_{i}$ passes through
$\pi$. However, a pair of geodesic rays meeting at an angle $\pi$ is also a
minimizing geodesic, so the transition is in fact continuous.

The above process defines a continuous final family $f_{1}$. We now define the
family $f_{t}$ for times $0<t<1$. As before, we can define it in each disk
$B_{i}$ individually. First, suppose $i\not=0$. For any curve $f(x)$ in our
family, the arcs $f(x)\cap B_{i}$ are simple and disjoint. Extend them to a
continuous family of disjoint simple arcs sweeping out $B_{i}$ and apply the
modified disk flow to these arcs individually. The resulting arcs are also
simple and disjoint. Therefore this gives rise to a diffeomorphism of the disk
that fixes the boundary. For $B_{1}$, we have to work a little harder to
produce a diffeomorphism because the .modified disk flow can map disjoint arcs
to arcs that intersect at $p$ and vice versa. However, by Lemma 3 our family
is formed from $p$-admissible curves both before and after applying the
modified disk flow. Therefore we can continuously perturb any curve and its
image inside $B_{1}$ so that they both consist of simple, disjoint arcs. Then
we can once again extend these arcs to a sweepout of $B_{i}$ and obtain a
diffeomorphism for each curve.

These diffeomorphisms depend continuously on the curve $f(x)$, since the image
by the modified disk flow depends continuously on the original curve by Lemma
9. Thus we have a continuous map of the parameter space $X$ into
$\operatorname{Diff}(D,\operatorname{rel}\partial)$, the space of boundary-
fixing diffeomorphisms of the disk. Because
$\operatorname{Diff}(D,\operatorname{rel}\partial)$ is contractible, this map
is homotopic to the constant map that sends every point of $X$ to the identity
diffeomorphism. Call this homotopy
$F:X\times[0,1]\to\operatorname{Diff}(D,\operatorname{rel}\partial)$. We thus
obtain a homotopy from the image family $f_{1}$ to the original family
$f=f_{0}$ by the map $f_{t}(x)\to(F(x,t))(f(x))$ for $x\in X,t\in[0,1]$, which
completes our definition of the modified disk flow on families. ∎

###  2.2 Short Meridional Slicings

We now need to find suitable families of curves to which we can apply the
modified disk flow. In the standard Lusternik–Schnirelmann proof, one can use
cycles made of meridians on the round sphere. While we could simply take the
images of those curves under a diffeomorphism, this would not provide us with
a length bound. Instead, we do the following. If $p$ is the point where we
want to base our geodesic loops, pick a point $q$ that is at locally maximal
distance from $p$ in $M$. These two points are analogous to the poles on the
sphere. By Berger’s Lemma, for any $v\in T_{q}M$ there is a minimizing
geodesic $\gamma$ from $q$ to $p$ with $\langle\gamma^{\prime}(0),v\rangle\geq
0$. Therefore we can pick either two or three such geodesics, denoted
$\tau_{i}$ and cyclically indexed, such that
$\langle\tau_{i}^{\prime}(0),\tau_{i+1}^{\prime}(0)\geq 0$. Our meridional
curves will be given by interpolating between these geodesics.

We will construct these interpolations in two steps. First, we will contract
the curves $-\tau_{i}*\tau_{i+1}$ through loops based at $p$, creating
homotopies whose lengths are bounded in terms of the diameter $d$. We then
apply the following lemma, due to M. Maeda [6], to convert this homotopy to a
homotopy between $\tau_{i}$ and $\tau_{i+1}$ through curves whose lengths are
also bounded.

######  Lemma 10 (M. Maeda, 1994 [6]).

Suppose $\gamma_{1}$ and $\gamma_{2}$ are arcs with shared endpoints $p$ and
$q$. Suppose also that $\gamma_{1}*-\gamma_{2}$ can be homotoped to $p$ via
loops based at $p$ of length at most $L$. Then $\gamma_{1}$ can be homotoped
to $\gamma_{2}$ through curves of length at most $L(\gamma_{1})+L$.∎

If the homotopy from $\gamma_{1}*-\gamma_{2}$ to $p$ given in the statement of
the lemma is denoted by $\Gamma_{t}$, then the homotopy from $\gamma_{1}$ to
$\gamma_{2}$ is given explicitly by applying the homotopy
$-\Gamma_{t}*\gamma_{1}$, $t\in[0,1]$, followed by the homotopy
$\gamma_{2}*-\gamma_{1}\mid_{[0,1-t]}*\gamma_{1}\mid_{[0,1-t]}$, $t\in[0,1]$.

First, however, we need to determine how to contract the loops
$-\tau_{i}*\tau_{i+1}$. If we use the modified disk flow, fixing $p$, the
homotopy may stall at a union of geodesic loops instead of a point curve.
Indeed, it is possible there are only two geodesics, $\tau_{1},\tau_{2}$,
which cover two halves of a single closed geodesic through $p$. Such a curve
will be fixed by the disk flow. Therefore a more involved algorithm is needed.

For brevity, we utilize the following definition.

######  Definition 2 (Geodesic Digons).

A closed region bounded by a closed curve consisting of the concatenation of
two (minimizing) geodesic segments is called a (minimizing) geodesic digon.
The two endpoints of the segments are called vertices.

In particular, the loops we want to contract are the boundaries of minimizing
geodesic digons with vertices at $p$ and $q$, and with an acute angle at $q$.
The behaviour of the angles at $p$ and $q$ will help us control the behaviour
of the modified disk flow.

We will also use the fact that $q$ lies in the cut locus of $p$, allowing us
to exploit the structure of the cut locus. In particular, we can take
advantage of the following theorem.

######  Theorem 2 (S. B. Myers, 1935 [7]).

Suppose $M$ is a Riemannian 2-sphere with an analytic metric. Then for any
$x\in M$, the cut locus of $x$ consists of either a single point or a finite
embedded tree. If the cut locus is a single point, this point is conjugate on
every geodesic connecting it to $x$. If instead the cut locus is a tree, then
the degree of any point $y$ equals the number of minimizing geodesics
connecting $x$ to $y$. Moreover, $M$ consists of a disk with geodesic polar
co-ordinates centred at $x$ glued along the cut locus of $x$. ∎

Because the cut locus is finite, we will be able to induct on the number of
vertices, say $v$, that are contained in our digon, which we will call
$\Delta$. Using Theorem 2, we will show that either $q$ is at a vertex of the
cut locus of $p$, or we can homotope $\Delta$ to a digon that contains at most
$v-1$ vertices of the cut locus. In the latter case, we proceed by induction
on $v$. If $q$ is at a vertex, then some number of edges of the cut locus
incident to $q$ lie in $\Delta$ (or else $\Delta$ would be easily contractible
to $p$). The number of these branches corresponds to the number of minimizing
geodesic digons with vertices at $p$ and $q$ that $\Delta$ covers. We will
choose a suitable “interior” digon, called $\Delta^{\prime}$, with the
property that the angle of $\Delta^{\prime}$ at $q$ is still at most $\pi$. We
then apply the modified disk flow to the boundary of $\Delta^{\prime}$. By the
following lemma, any loops obtained from applying the modified disk flow lie
within $\Delta^{\prime}$.

######  Lemma 11.

Suppose $\Omega$ is a closed region bounded by a piecewise geodesic curve
$\partial\Omega$ that passes through $p$. Suppose every interior angle at
every vertex of $\partial\Omega$, except perhaps the one at $p$, is at most
$\pi$. Then any simple curve $\gamma$ in $\Omega$ (including $\partial\Omega$
if it is simple) remains in $\Omega$ under the modified disk flow.

###### Proof.

Let the family obtained by applying the modified disk flow to $\gamma$ be
denoted by $\gamma_{t}$. We need to ensure that for every $t$, $\gamma_{t}$
lies within $\Omega$. It is enough to check that $\gamma$ remains in $\Omega$
after applying the disk flow once. In each ball $B_{i}$ with $i>1$,
$\Omega\cap B_{i}$ is a convex region containing $\gamma\cap B_{i}$. Therefore
when each arc of $\gamma\cap B_{i}$ is replaced by a minimizing geodesic
segment, this segment will necessarily lie in $\Omega\cap B_{i}$.

Now consider $B_{1}$. If the interior angle of $\partial\Omega$ at $p$ is at
most $\pi$, then $\Omega\cap B_{1}$ is also convex and the result holds.
Suppose instead the angle at $p$ is greater than $\pi$. As before, let
$\gamma_{0}$ denote the arc of $\gamma\cap B_{1}$ through $\gamma(0)$ and
denote the two rays connecting $p$ to the endpoints of $\gamma_{0}$ by
$\alpha$ and $\beta$. Consider the closed sector $S$ in $\Omega\cap B_{1}$
bounded by $\alpha$ and $\beta$ (since $p\in\partial\Omega$, there is a unique
such sector). Then $(\Omega\cap B_{1})\setminus S$ consists of two other
sectors $S^{\prime}$ and $S^{\prime\prime}$ with vertex at $p$. Suppose there
is another arc of $\gamma\cap B_{1}$, say $\gamma^{\prime}$. Because $\gamma$
is simple, $\gamma^{\prime}$ does not pass through $p$ and hence must lie in
exactly one of the three sectors. As in Lemma 2, $\gamma^{\prime}$ bounds a
fourth sector with vertex at $p$ that does not contain the endpoints of
$\gamma_{0}$, and $\gamma^{\prime}$ is only replaced by a minimizing geodesic
if that sector has an acute angle at $p$ (i.e., if it is convex). If $\gamma$
is replaced by two geodesic rays through $p$, then its image under the
modified disk flow will necessarily lie in $\Omega$. Otherwise, the fourth
sector has an acute angle. Because $\gamma^{\prime}$ does not pass through
$p$, this sector is strictly contained in either $S,S^{\prime}$ or
$S^{\prime\prime}$. Therefore $\gamma^{\prime}$ lies in a convex sector within
$\Omega$, and hence the minimizing geodesic connecting the endpoints of
$\gamma^{\prime}$ lies entirely within $\Omega\cap B_{1}$, as desired. ∎

Therefore we either succeed in contracting $\partial\Delta^{\prime}$ or we
encounter some simple geodesic loops in $\Delta^{\prime}$ based at $p$ of
length at most $L(\partial\Delta^{\prime})\leq 2d$. If $M$ contains two such
loops, then Theorem 1 is proved. Therefore we can assume that either we can
contract $\partial\Delta^{\prime}$ or $\Delta^{\prime}$ contains the only such
loop. Then any digon satisfying the hypotheses of Lemma 11 that lies outside
of $\Delta^{\prime}$ is contractible via the disk flow. Therefore if we can
contract the loop in $\Delta^{\prime}$, we can homotope $\Delta$ to
$\Delta\setminus\Delta^{\prime}$ via Lemma 10. Then
$\Delta\setminus\Delta^{\prime}$, and hence $\Delta$ itself, is easily
contractible. We detail this process in the following lemma.

######  Lemma 12.

Let $\Delta$ be a minimizing geodesic digon with vertices $p$ and $q$. Suppose
also the interior angle of $\partial\Delta$ at $q$ is at most $\pi$. Then
either there are at least two simple geodesic loops at $p$ of length at most
$2d$ in the interior of $\Delta$ or $\partial\Delta$ can be contracted to $p$
through loops based at $p$ of length at most $6d$.

###### Proof.

Assume that $\Delta$ contains at most one such loop. Recall by Theorem 2 that
the cut locus of $p$ has the structure of an embedded tree or a single point.
We will consider a point to be a tree consisting of a single vertex.
Therefore, suppose that the closure of $\Delta$ contains exactly $v$ vertices
of the cut locus of $p$. We will proceed by induction on $v$ and the number of
loops contained in $\Delta$, which we denote by $m$, to show that $\Delta$ can
be contracted through loops based at $p$ of length bounded by some function
$c(v,m)$. Note that $v\geq 1$, since $q$ lies on the cut locus of $p$, and
$0\leq m\leq 1$ by assumption.

Note that if $q$ lies within an edge of the cut locus (i.e., it is not a
vertex), then $\partial\Delta$ can be homotoped to a minimizing geodesic digon
with vertices at $p$ and some cut locus vertex $q^{\prime}\in\Delta$. This is
possible since minimizing geodesics vary continuously as their endpoint moves
within an edge of the cut locus. Thus we can assume that $q$ lies on a vertex.

Our base case is when $v=1$. Since $q$ is a vertex, the cut locus intersects
$\partial\Delta$ only at $q$. Therefore $\partial\Delta$ can be contracted
through curves of length at most $c(1,m)=2d$ by perturbing it off of $q$ and
applying the modified disk flow.

Suppose $v>1$. Note that the cut locus must be a tree in this case. Therefore
by Theorem 2, the total number of minimizing geodesic segments connect $p$ and
$q$ depends on the degree of $q$. By the same result, at least one edge of the
cut locus incident to $q$ is contained in $\Delta$. If only one edge lies in
$\Delta$, then $\Delta$ can be homotoped to a new digon (contained in
$\Delta$) by moving its two bounding geodesics along this edge until they
reach the next vertex contained in $\Delta$. This new digon contains $v-1$
vertices, so by induction it can be contracted through loops of length
$c(v-1,m)$.

Suppose instead that there are at least two edges of the cut locus incident to
$q$ within $\Delta$. Then the geodesics emanating from between these edges
split $\Delta$ into minimizing geodesic digons whose interiors are disjoint.
Exactly two of these are bounded by one of the edges of $\partial\Delta$ and
some other minimizing geodesic. Because the angles between adjacent minimizing
geodesics at each vertex must sum to $2\pi$, at least one of these two digons
has an angle at $q$ that is at most $\pi$. Call this digon $\Delta^{\prime}$.
Apply the modified disk flow to $\partial\Delta^{\prime}$.

If $\partial\Delta^{\prime}$ contracts to a point, then we can apply Lemma 10
to homotope one of the edges of $\partial\Delta^{\prime}$ to the other via
curves of length at most $3d$. Therefore $\partial\Delta$ can be homotoped to
$\partial(\Delta\setminus\Delta^{\prime})$ though curves of length at most
$4d$, since $\partial\Delta^{\prime}$ and $\partial\Delta$ share an edge.
Because $\Delta^{\prime}$ contains an edge of the cut locus and hence at least
one vertex in its interior, $\Delta\setminus\Delta^{\prime}$ contains at most
$v-1$ vertices of the cut locus. Thus by induction this digon can be
contracted through loops of length $c(v-1,m)$. Thus $\Delta$ is contractible
through loops of length at most $\max\\{4d,c(v-1,m)\\}$.

If $m=1$, then the other possibility is that the modified disk flow will
contract $\partial\Delta^{\prime}$ to the unique simple geodesic loop based at
$p$ of length at most $2d$, which we will call $\gamma$. We need to contract
this loop in order to finish our contraction of $\Delta$. Because $\gamma$ is
a geodesic loop (and hence minimizing for at most half its length), it must
also intersect the cut locus of $p$. Follow $\gamma$ from $\gamma(0)$ to its
first intersection point with the cut locus, say $\gamma(t_{0})$. By
definition, the arc $\gamma\mid_{[0,t_{0}]}$ is minimizing. If
$\gamma\mid_{[t_{0},1]}$ is also minimizing, then $\gamma$ itself bounds a
minimizing geodesic digon in $\Delta^{\prime}$ with vertices at $p$ and
$\gamma(t_{0})$. Moreover, this digon has angle $\pi$ at $\gamma(t_{0})$ and
does not contain any short geodesic loops in its interior. Therefore by
induction we can contract $\gamma$ through curves of length at most
$c(v-1,0)$. Consequently, by Lemma 10 $\partial\Delta$ can be homotoped to
$\partial(\Delta\setminus\Delta^{\prime})$ though curves of length at most
$2d+c(v-1,0)$, and then $\Delta\setminus\Delta^{\prime}$ can be contracted
through curves of length at most $c(v-1,0)$.

On the other hand, suppose $\gamma\mid_{[t_{0},1]}$ is not minimizing. If
$\gamma(t_{0})$ is a leaf vertex, then as before we can contract $\gamma$
through curves of length at most $2d$. Otherwise, there is some other
minimizing geodesic $\eta$ connecting $p$ to $\gamma(t_{0})$. We claim that
$\eta$ splits the region enclosed by $\gamma$ into two digons (see Figure 10,
left). Because $\eta$ is a minimizing geodesic, the only other potential
possibility is that $\eta$ lies outside the region enclosed by $\gamma$ but
inside $\Delta^{\prime}$ (see Figure 10, right). In this case, consider the
region bounded by $\eta$, $\gamma\mid_{[t_{0},1]}$ and $\partial\Delta$. This
region is bounded by a piecewise geodesic with all interior angles acute
except perhaps for one of the angles at $p$. Therefore by Lemma 11 the image
of $\partial\Delta^{\prime}$ under the modified disk flow must have remained
in this region. This contradicts the fact that the image converged to
$\gamma$. Thus $\eta$ must split the region bounded by $\gamma$ into two
digons. The angles formed by this pair of digons at $\gamma(t_{0})$ are both
acute since they lie within the region bounded by $\gamma$. Moreover, neither
of these digons contain a simple geodesic loop at $p$ of length at most $2d$.
Therefore by Lemma 11 the boundaries of each digon can be shortened to a point
through loops of length at most $2d$. As before, we can then contract $\gamma$
(and hence $\Delta^{\prime}$) through loops of length at most $4d$ and then
homotope $\partial\Delta$ to $\partial(\Delta\setminus\Delta^{\prime})$ though
loops of length at most $6d$. Then $\Delta\setminus\Delta^{\prime}$ can be
contracted via curves of length at most $c(v-1,0)$.

In summary, we have $c(1,0)=2d$ and $c(v,0)=\max\\{4d,c(v-1,0)\\}$, and hence
$c(v,0)=4d$. Moreover, $c(1,1)=2d$ and

$\displaystyle
c(v,1)=\max\\{6d,c(v-1,1),2d+c(v-1,0)\\}=\max\\{c(v-1,1),6d\\}.$

Thus we obtain $c(v,m)=6d$. ∎

$p$$\gamma$$\gamma(t_{0})$$\eta$$\Delta^{\prime}$$\Delta^{\prime}$$p$$\gamma$$\gamma(t_{0})$$\eta$$\Delta^{\prime}$
Figure 10: The two possible ways a minimizing geodesic $\eta$ (dashed) can
intersect the geodesic loop $\gamma$ within the digon $\Delta^{\prime}$.

Now we know how to contract digons, and hence we know how to contact the disks
we obtain by applying Berger’s lemma. Therefore we are almost ready to
construct our meridional slicing. We will be forming our curve families from
pairs of meridional curves, and we need the resultant curves to be simple. We
will do this by ensuring that our meridional slicing is “monotone”.

######  Definition 3 (Monotone Slicings).

We call a meridional slicing $\Gamma_{t}$ monotone if the curves $\Gamma_{t}$
are simple and $\Gamma_{t}\cap\Gamma_{s}=\\{p,q\\}$ for $s\not=t$.

The existence of a monotone meridional slicing with endpoints at $p$ and $q$
is given by the following three lemmas, the first of which is presented
without proof.

######  Lemma 13.

Let $\gamma$ be a simple piecewise geodesic curve. Let the family of curves
given by applying the modified disk flow to $\gamma$ be denoted $\gamma_{t}$.
Then $\gamma_{t}$ is homotopic to a monotone homotopy through $p$-admissible
curves starting at $\gamma$ and ending at $\gamma_{1}$. Moreover, the image of
the new family lies within the image of the original family.∎

######  Lemma 14.

The contracting homotopy described in Lemma 12 can be made monotone.

###### Proof.

The homotopy in Lemma 12 is given by a concatenation of homotopies of digons
and geodesic loops. Because any two minimizing geodesics emanating from $p$ do
not intersect except at $p$ and perhaps a point on the cut locus, the homotopy
formed by moving a pair of minimizing geodesics down an edge of the cut locus
of $p$ is monotone. Moreover, the image of the homotopy lies between the
digons formed at the vertices of the edge. Therefore it is enough to check
that when shortening a digon in our process, the resulting homotopy is
monotone and its image lies within the digon. This holds by Lemmas 11 and 13.
∎

Now we combine the results of this section to form our meridional slicing.

######  Lemma 15.

Given $p\in M$, pick any $q$ at locally maximum distance from $p$ in $M$.
Suppose there is at most one geodesic loop based at $p$ of length at most $2d$
in $M$. Then for any $\epsilon>0$ there is a monotone meridional slicing
$\Gamma_{t}$ of $M$ through curves with endpoints at $p$ and $q$ of length at
most $7d+\epsilon$.

###### Proof.

First we find two or three minimizing digons with vertices at $p$ and $q$ as
detailed in the introduction to this section. Because $q$ is at locally
maximal distance from $p$, Berger’s lemma ensures that given any $v\in T_{q}M$
there is a minimizing geodesic $\gamma$ connecting $q$ to $p$ with
$\langle\gamma^{\prime}(0),v\rangle\geq 0$. In particular, there are either
two or three minimizing geodesics $\tau_{i}$ from $q$ to $p$ of length at most
$d$, ordered cyclically, so that each pair $\tau_{i}*-\tau_{i+1}$ bounds a
disk $D_{i}$ and the closure of these disks cover $M$. Moreover, because the
angle formed at $q$ between each adjacent pair is at most $\pi$, at least two
of these disks must be convex. If there is a third disk it may have an obtuse
angle at $p$. Without loss of generality, order the disks so that $D_{1}$ and
$D_{2}$ are both convex.

Contract each $\partial D_{i}=-\tau_{i}*\tau_{i+1}$ through loops based at $p$
via Lemma 12, obtaining a homotopy $H^{i}_{t}$ with $L(H^{i}_{t})\leq 6d$. By
Lemma 14, we can assume that this homotopy is monotone. Moreover, the curves
$H^{i}_{t}$ are contained in $D_{i}$ by Lemma 11. We now use these homotopies
to construct a meridional slicing of $M$, although we will have to be careful
to maintain monotonicity. One way to homotope $\tau_{i}$ to $\tau_{i+1}$ is to
follow the curves $\tau_{i}*H_{t}^{i}$ from $\tau_{i}=\tau_{i}*\\{p\\}$ to
$\tau_{i}*-\tau_{i}*\tau_{i+1}$, and then contract $\tau_{i}*-\tau_{i}$ along
itself as in Lemma 10. However, we need to alter this homotopy to be monotone.

Pick $\epsilon>0$ small enough that $B_{\epsilon}(p)$ is a totally normal
metric ball. Let $f^{i}(t)$ be the first time that $H_{t}^{i}$ intersects
$\partial B_{\epsilon(t+1)/2}(p)$, so that
$P_{t}^{i}=H_{t}^{i}\mid_{[f^{i}(t),1]}$ is a curve from some point on
$\partial B_{\epsilon(t+1)/2}(p)$ to $p$. The curve $P^{i}_{t}$ will form the
second half of the meridian $\Gamma_{t}$. The first half will be given by a
curve $\sigma_{t}^{i}$ from $q$ to $H_{t}^{i}(f^{i}(t))$, defined so that the
curves $\sigma_{t}^{i}$ do not mutually intersect except at $q$ and do not
intersect any $H_{s}^{i}$ (except perhaps at their endpoints if $s=t$).

First we define a curve $T_{t}^{i}$ from $q$ to $\partial B_{\epsilon}(p)$,
which will form the first portion of $\sigma_{t}^{i}$. Because $\tau_{i}$ is
minimizing between $p$ and $q$, $\tau_{i}\cap\partial B_{\epsilon}(p)$ does
not lie on the cut locus of $q$. Therefore, since the cut locus of $q$ is a
tree (or a mere point), there is some arc $\Sigma^{i}$ of $\partial
B_{\epsilon}(p)\cap D_{i}$ starting at $\tau_{i}\cap\partial B_{\epsilon}(p)$
that does not lie in the cut locus of $q$. Parameterize this arc by $[0,1]$
and let $T_{t}^{i}$ be the minimizing geodesic connecting $q$ to
$\Sigma^{i}(t)$, ordered so that $T_{0}^{i}$ is an arc of $\tau_{i}$ (see
Figure 11, left). Necessarily, the curves $T_{t}^{i}$ are simple, lie in
$D_{i}\setminus B_{\epsilon}(p)$ and do not mutually intersect except at $q$.

Now we connect $T_{t}^{i}$ to $H_{t}^{i}(f^{i}(t))$. Let
$\rho^{i}_{t}:[0,1]\to M$ be a portion of a radial geodesic in
$B_{\epsilon}(p)$ that connects $T_{t}^{i}(1)\in\partial B_{\epsilon}(p)$ to a
point on $\partial B_{\epsilon(t+1)/2}(p)$ (see Figure 11, centre). Let
$\eta^{i}_{t}:[0,1]\to M$ be the arc of $\partial B_{\epsilon(t+1)/2}(p)\cap
D_{i}$ that starts at $\rho^{i}_{t}(1)$ and ends at the first point of
$\partial B_{\epsilon(t+1)/2}(p)\cap H_{t}^{i}$ it meets (see Figure 11,
right). The curve $\sigma_{t}^{i}$ is then defined as
$\sigma^{i}_{t}=T_{t}^{i}*\rho_{t}^{i}*\eta_{t}^{i}$. We stress that, again,
these curves are simple and do not mutually intersect except at their
endpoints.

We are now ready to build the portion of $\Gamma_{t}$ given by homotoping
$\tau_{i}$ to $\tau_{i+1}$. Recall that
$P^{i}_{t}=H_{t}^{i}\mid_{[f^{i}(t),1]}$ and fix a small $\delta>0$. First,
homotope $\tau_{i}$ to $\sigma^{i}_{\delta}*P^{i}_{\delta}$ via some monotone
homotopy, choosing $\delta$ and $\epsilon$ so that this homotopy passes
through reasonably short curves. Then use the homotopy
$t\mapsto\sigma^{i}_{t}*P^{i}_{t}$ to pass from
$\sigma^{i}_{\delta}*P^{i}_{\delta}$ to
$\sigma^{i}_{1-\delta}*P^{i}_{1-\delta}$. For small enough $\delta$ and
$\epsilon$, the curve $\sigma^{i}_{1-\delta}*P^{i}_{1-\delta}$ is very close
to $\tau_{i}*-\tau_{i}*\tau_{i+1}$. Therefore we can apply some monotone
homotopy through reasonably short curves to contract one “arm” of
$\sigma^{i}_{1-\delta}*P^{i}_{1-\delta}$ (the portion approximating
$\tau_{i}*-\tau_{i})$, leaving a curve that lies very close to $\tau_{i+1}$.
We then homotope this curve to $\tau_{i+1}$. Our desired monotone meridional
slicing $\Gamma_{t}$ is given by using these homotopies to pass from
$\tau_{1}$ to $\tau_{2}$, then to $\tau_{3}$ if it exists, and then back to
$\tau_{1}$. Given any constant $c>0$, we can choose $\epsilon$ and $\delta$
small enough that the curves $\Gamma_{t}$ are bounded in length by $d+6d+c$.
Moreover, because the homotopies between $\tau_{i}$ and $\tau_{i+1}$ are
monotone and lie in disjoint disks $D_{i}$, a generic point on $M$ lies in
exactly one curve in this family. Thus this family represents a degree $\pm 1$
map into $M$, and hence is indeed a meridional slicing. ∎

$B_{\epsilon}(p)$$p$$\tau^{i}$$T^{i}_{s}$$T^{i}_{1}$$q$$B_{\epsilon}(p)$$B_{\epsilon(t+1)/2}(p)$$T^{i}_{t}$$H^{i}_{t}$$\rho_{t}^{i}$$\eta_{t}^{i}$
Figure 11: Left: Defining $T_{t}^{i}$. Centre: Defining $\rho_{t}^{i}$. Right:
Defining $\eta_{t}^{i}$. The solid line is a typical curve in the meridional
slicing.

###  2.3 Counting Loops

We will now use the modified disk flow to prove the existence of two short
simple geodesic loops at $p$ in analogy with the proofs of the
Lusternik–Schnirelmann theorem given in [3, 4].

###### Proof of Theorem 1.

Because we want our geodesic loops to have distinct images, we would like to
identify any two curves that differ only in parameterization or orientation.
However, the modified disk flow depends on a curve’s parameterization –
specifically, on the choice of which arc through $p$ contains the image of
$t=0$. Therefore we will follow [4] and consider both parameterized and
unparameterized curves in our proof. Let $\Pi_{p}M$ be the space of
parameterized, oriented $p$-admissible curves in $M$. Let $\pi$ be the
projection map that sends a parameterized curve to the corresponding
unparameterized curve and define $\Sigma_{p}M=\pi(\Pi_{p}M)$. We will also
make use of the orientation reversing map $R:\Pi_{p}M\to\Pi_{p}M$ defined by
$R(\gamma)=-\gamma$.

We are interested in chains in $\Pi_{p}M$ and cycles in $\Sigma_{p}M$ with
$\mathbb{Z}_{2}$ coefficients. If $M$ does not admit two simple geodesic loops
at $p$ of length at most $2d$, then by Lemma 15 we have a monotone meridional
slicing $\Gamma_{t}$, of $M$ through curves starting at $p$ and ending at some
$q$. Moreover, $L(\Gamma_{t})$ is at most $7d$ plus some arbitrarily small
constant $c>0$. We will use $\Gamma_{t}$ to define two chains in $\Pi_{p}M$
which we call $u_{0}$ and $u_{1}$. Our cycles in $\Sigma_{p}M$ will be the
images under $\pi$ of these chains.

For convenience, we will extend $\Gamma_{t}$ periodically to all
$t\in\mathbb{R}$. Define $u_{0}:([0,1],\\{0,1\\})\to(\Pi_{p}M,\\{p\\})$ as
follows. Fix some small $\epsilon>0$. For $t\in[\epsilon,1-\epsilon]$, take
$u_{0}(t)=\Gamma_{t}*-\Gamma_{-t}$. For $t\in[0,\epsilon)$, let $u_{0}$ be a
monotone homotopy from $p$ to $\Gamma_{\epsilon}*-\Gamma_{-\epsilon}$ that
increases lengths by at most some small $\delta$, where we take $\epsilon>0$
small enough that this is possible. Similarly, for $t\in(1-\epsilon,1]$, let
$u_{0}$ be a monotone homotopy from
$\Gamma_{1-\epsilon}*-\Gamma_{-1+\epsilon}$ to $p$ that increases lengths by
at most $\delta$. The second map
$u_{1}:([0,1],\\{0,1\\})\times[0,1/2]\to(\Pi_{p}M,\\{p\\})$ is essentially
given by rotating $u_{0}$ through angle $\pi$. Specifically, for
$(t,s)\in[\epsilon,1-\epsilon]\times[0,1/2]$, take
$u_{1}(t,s)=\Gamma_{s+t}*-\Gamma_{-s-t}$. When $t\in[0,\epsilon)$, let $u_{1}$
be a monotone homotopy from $p$ to $\Gamma_{s+\epsilon}*-\Gamma_{-s-\epsilon}$
that increases lengths by at most $\delta$, and when $t\in(1-\epsilon,1]$ let
$u_{1}$ be a monotone homotopy from
$\Gamma_{s+1-\epsilon}*-\Gamma_{-s-1+\epsilon}$ to $p$. Moreover, we define
both homotopies in a way that makes $u_{1}$ continuous. Finally, for
notational convenience we reparameterize $u_{0}$ and $u_{1}$ to have domains
$t\in[-1,1]$ and $(t,s)\in[-1,1]\times[0,1]$ respectively.

Define the $\mathbb{Z}_{2}$-cycles $v_{i}=\pi(u_{i})$. We are concerned with
the following two classes of cycles in $\Sigma_{p}M$. First, let $V_{0}$ be
the smallest class of $\mathbb{Z}_{2}$-cycles of the form $v:([-1,1],\\{\pm
1\\})\to(\Sigma_{p}M,\\{p\\})$ that contains $v_{0}$ and contains all
$\mathbb{Z}_{2}$-cycles that are homotopic to $v_{0}$ through
$\mathbb{Z}_{2}$-cycles. By this we mean if $v\in V_{0}$ then there exists a
continuous map $h:([-1,1],\\{\pm 1\\})\times[0,1]\to(\Pi_{p}M,\\{p\\})$ such
that $h(\cdot,0)=u_{0}$ and $v=\pi(h(\cdot,1))$. Similarly, $V_{1}$ is the
smallest class of cycles $v:([-1,1],\\{\pm
1\\})\times[0,1]\to(\Sigma_{p}M,\\{p\\})$ that contains $v_{1}$ and contains
all $\mathbb{Z}_{2}$-cycles $v$ that are homotopic to $v_{1}$, in the sense
that there is a map $h:([-1,1],\\{\pm
1\\})\times[0,1]^{2}\to(\Pi_{p}M,\\{p\\})$ such that $h(\cdot,\cdot,0)=u_{1}$,
$v=\pi(h(\cdot,\cdot,1))$, and $h(t,0,\cdot)=R(h(-t,1,\cdot))$ for each
$t\in[-1,1]$.

We want to find geodesic loops whose lengths realize the min-max values
$l_{i}=\min_{v\in V_{i}}\max_{\tau}\\{L(v(\tau))\\}$. Note that $l_{0}\leq
l_{1}$ because $u_{0}(t)=u_{1}(t,0)$ and hence any homotopy of $v_{1}$
determines a homotopy of $v_{0}$. Secondly, we claim that $l_{0}>0$. Let
$f:S^{2}\to M$ be any map that is not nullhomotopic (e.g., the identity map).
Suppose $u_{0}^{\tau}$ is a homotopy of $u_{0}$ in the above sense. We want to
parameterize $S^{2}$ by $u_{0}$ and then convert this homotopy to a homotopy
$f^{\tau}$ of $f$. Every point in $S^{2}$ lies on some curve $u_{0}(t)$.
First, perturb $u_{0}$ slightly near $q$ so that only one curve crosses $q$
and that all curves intersect only at $p$. Then for $x\not=p$ the choice of
curve containing $x$ is unique, so if $x=(u_{0}(t))(s)$, we define
$f^{\tau}(x)=u_{0}^{\tau}(t)(s)$. Our map is also well-defined on $p$ via
$f^{\tau}(p)=p$. Now suppose that $l_{0}=0$, so that $u_{0}$ can be homotoped
so that its image lies in an arbitrarily small disk around $p$. Then $u_{0}$
can be homotoped via some $u_{0}^{\tau}$ to the constant map $p$, and hence
$f$ can be homotoped via some $f^{\tau}$ to the constant map $p$. This is a
contradiction of the fact that $f$ is not nullhomotopic, and hence we see that
$l_{0}>0$.

In addition, our definition of $u_{0}$ and $u_{1}$ shows that $l_{i}\leq
2\max_{t}\\{L(\Gamma_{t})\\}=14d+\delta$ for any $\delta>0$, and hence
$l_{i}\leq 14d$. In fact, $l_{0}\leq 8d$. This is because $V_{1}$ contains a
cycle defined similarly to $v_{0}$ except using $\Gamma_{0}*-\Gamma_{-t}$ in
place of $\Gamma_{t}*-\Gamma_{t}$, where without loss of generality
$\Gamma_{0}$ is a shortest curve in the family $\Gamma_{t}$ (and hence of
length at most $d$ by construction).

Before we can prove anything else useful about $l_{i}$ and any corresponding
critical points, we need the following intermediate lemma (c.f. Lemma 3.12 in
[3]).

######  Lemma 16.

Let $K$ be the (possibly empty) set of unions of non-trivial simple geodesic
loops at $p$ of total length $l>0$. Given any open neighbourhood $W$ of $K$ in
$\Pi_{p}M$, there exists $\delta>0$ such that for any $p$-admissible curve
$\gamma$ with $L(\gamma)<l+\delta$, either the modified disk flow decreases
the length of $\gamma$ by at least $2\delta$ or $\gamma\in W$.

###### Proof.

Suppose the conclusion did not hold. Then we could find some $p$-admissible
loops $\gamma_{m}$ so that

  1. 1.

$L(\gamma_{m})<l+1/m$

  2. 2.

$L(\gamma_{m}^{\prime})>l-1/m$

  3. 3.

$\gamma_{m}\not\in W$

where $\gamma_{m}^{\prime}$ is obtained by applying the modified disk flow to
$\gamma_{m}$ once. Because their lengths are uniformly bounded, there is a
subsequence of the $\gamma_{m}$ that converges to some $\gamma_{\infty}$. As
in the proof of Lemma 5, the modified disk flow must fix the length of
$\gamma_{\infty}$, or else for some large $m$ we would be able to contradict
the assertion that $L(\gamma_{m}^{\prime})>l-1/m$. Therefore by Lemma 5,
$\gamma_{\infty}$ is either a point curve or a union of simple geodesic loops
at $p$. Moreover, by the same argument we must have
$L(\gamma_{\infty})=\lim\limits_{m\to\infty}L(\gamma_{m})=l$, since if
$\gamma_{\infty}$ was much shorter than the $\gamma_{m}$ for large $m$, then
$\gamma^{\prime}_{m}$ is much shorter than $\gamma_{m}$. Note that this means
that $\gamma_{\infty}$ cannot be a point curve, because in this case we could
find some $\gamma_{m}$ of arbitrarily small length, and hence some
$\gamma_{m}^{\prime}$ of arbitrarily small length. Therefore $\gamma_{\infty}$
must be a union of simple geodesic loops, so since $L(\gamma_{\infty})=l$,
$\gamma_{\infty}$ is an element of $K$. In particular, $\gamma_{\infty}\in W$
and hence $\gamma_{m}\in W$ for sufficiently large $m$. This is a
contradiction of our choice of $\gamma_{m}$. ∎

Using this lemma, we will show that there exists a union of simple geodesic
loops at $p$ with collective length realizing the min-max value $l_{i}$.
Suppose there was no such set of loops. Define $\Pi_{p}^{l}M$ as the set of
elements of $\Pi_{p}M$ of length at most $l$. Then by Lemma 16, there is some
$\epsilon>0$ such that the modified disk flow maps $\Pi_{p}^{l_{i}+\epsilon}M$
into $\Pi_{p}^{l_{i}-\epsilon}M$ (since we can take $W=\emptyset$). Consider a
cycle $v\in V_{i}$ whose image consists of curves of length at most
$l_{i}+\epsilon$. Then by the definition of $V_{i}$ there is a continuous map
$h$ such that $h(\cdot,0)=u_{i}$, $v=\pi(h(\cdot,1))$, and each $h(\cdot,t)$
is a $\mathbb{Z}_{2}$-cycle in $\Pi_{p}M$. Define $u=h(\cdot,1)$. Then by
Lemma LABEL:lemma:family_homotopic $u$ will be mapped under the modified disk
flow to a homotopic map $u^{\prime}$ whose image consists of curves of length
at most $l_{i}-\epsilon$. However, because $\pi(u)\in V_{i}$ we have
$\pi(u^{\prime})\in V_{i}$. This is because the outcome of the modified disk
flow homotopy does not depend on the orientation of the curve it is applied
to, and point curves are fixed, so the homotopy from $u$ to $u^{\prime}$
satisfies the requisite properties for $\pi(u^{\prime})$ to be in $V_{i}$.
This contradicts the definition of $l_{i}$, so the desired loops must exist.

Recall that we desire two distinct simple geodesic loops. If the union of
loops realizing the value $l_{0}$ contains at least two distinct loops in its
image, then there are two distinct simple geodesic loops at $p$ whose lengths
sum to at most $l_{0}$. Therefore the shorter has length at most $l_{0}/2$,
and the longer has length at most $l_{0}$. Otherwise, the critical value
$l_{0}$ is realized by a single prime simple geodesic loop of length $l_{0}$.
In this case we consider the union of loops realizing the value $l_{1}$.
Again, if there are at least two distinct loops in its image, then there are
two distinct simple geodesic loops at $p$ whose lengths are at most $l_{1}/2$
and $l_{1}$, although it is possible that one of these loops is the same as
the one that realizes the critical value $l_{0}$. The case of concern is when
both $l_{0}$ and $l_{1}$ are realized by a single loop. If $l_{0}\not=l_{1}$,
then these loops must be distinct (since they are prime) and we are done.
Suppose instead $l=l_{0}=l_{1}$ and we obtain identical prime loops of length
$l$. If there are any simple geodesic loops at $p$ of length strictly less
than $l$, again we are done because this loop must be distinct from the
(prime) loop of length $l$. If this is not the case, then we claim that there
are in fact infinitely many distinct simple geodesic loops at $p$ of length
$l$. We now prove this assertion.

Let $W$ be an open neighbourhood of the set of simple geodesic loops at $p$ of
length $l$. Note that this time we do not need to consider unions of geodesic
loops, since we have assumed there are no geodesic loops of length less than
$l$. Using Lemma 16, choose $\epsilon$ so that the modified disk flow sends
$\Pi^{l+\epsilon}_{p}M$ into $W\cup\Pi^{l-\epsilon}_{p}M$. Then we apply the
modified disk flow to a chain of curves of length at most $l+\epsilon$ that
projects into $V_{1}$, obtaining a map $u:([-1,1],\\{\pm
1\\})\times[0,1]\to(\Pi_{p}M,\\{p\\})$ such that $v=\pi(u)\in V_{1}$ and the
image of $u$ lies in $W\cup\Pi^{l-\epsilon}_{p}M$.

Pick a homotopically non-trivial curve $c(t)=(c_{1}(t),c_{2}(t))$ where
$c_{1}:[0,1]\to[-1,1]$ and $c_{2}:[0,1]\to[0,1]$ with
$c_{1}(0)=-c_{1}(1),c_{2}(0)=0$ and $c_{2}(1)=1$. We claim that
$\operatorname{img}c\cap u^{-1}(W)$ is non-empty. Because there are no simple
geodesic loops at $p$ of length strictly less than $l$, the modified disk flow
eventually maps every curve in $\Pi^{l-\epsilon}_{p}M$ to the point curve $p$.
Therefore if $\operatorname{img}c\cap u^{-1}(W)=\emptyset$, $u\circ c$ would
map into $\Pi^{l-\epsilon}_{p}M$ and hence we could use the modified disk flow
to homotope $u\circ c$ to the constant map $p$. Thus $v\circ c$ would be
nullhomologous. However, we also claim that $v\circ c$ is homologous to
$v(\cdot,0)$, which is not nullhomologous. Let
$H:[0,1]^{2}\to[-1,1]\times[0,1]$ be defined by

$H(s,t)=(s(c_{1}(t)+1)-1,c_{2}(t))$

We claim that $\partial(\pi(u\circ H))=v(\cdot,0)+v\circ c$, proving our
claim. Note that $H$ is just the straight-line homotopy between the curves
$(-1,c_{2}(t))$ and $c(t)$. Thus its boundary consists of the maps
$t\mapsto(-1,c_{2}(t)),t\mapsto c(t),s\mapsto(s(c_{1}(0)+1)-1,0)$ and
$s\mapsto(s(1-c_{1}(0))-1,1)$. Since $u(-1,\cdot)=p$, the image of
$t\mapsto(-1,c_{2}(t))$ under $u$ is the point curve $p$. Because
$u(t,0)=R(u(-t,1))$, the image of $s\mapsto(s(1-c_{1}(0))-1,1)$ under $s$ is
the same as the image under $u$ of $s\mapsto(s(c_{1}(0)-1)+1,0)$, up to
orientation. With $\mathbb{Z}_{2}$ coefficients, the chains
$u(s(c_{1}(0)-1)+1,0)$ and $u(s(c_{1}(0)+1)-1,0)$ sum to $u(\cdot,0)$.
Therefore $\partial(\pi(u\circ H))=v(\cdot,0)+v\circ c$, so $v(\cdot,0)$ is
homologous to $v\circ c$ in $\mathbb{Z}_{2}$ coefficients. This is a
contradiction, and hence $c\cap u^{-1}(W)\not=\emptyset$.

Consequently, in fact $u^{-1}(W)$ supports a $\mathbb{Z}_{2}$ 1-cycle, since
otherwise we can assume $\partial(u^{-1}(W))$ consists of closed curves and we
could concatenate arcs of $c\cap(u^{-1}(W))^{c}$ with arcs of
$\partial(u^{-1}(W))$ to form a closed curve that does not intersect
$u^{-1}(W)$. This is a contradiction unless this curve bounds a disk, in which
case the original curve can be pushed entirely into $u^{-1}(W)$. The 1-cycle
supported in $u^{-1}(W)$ is the image under $\pi$ of a map
$k(t)=(k_{1}(t),k_{2}(t))$, where $k_{1}:[0,1]\to[-1,1]$ and
$k_{2}:[0,1]\to[0,1]$ with $k_{1}(0)=-k_{1}(1),k_{2}(0)=0$ and $k_{2}(1)=1$.
If there are only finitely many simple geodesic loops at $p$ of length $l$,
then we can choose $W$ small enough that each connected component contains
only curves homotopic to a particular parameterized geodesic loop. Then the
image of $u\circ k$ lies entirely in one such connected component. However,
the curves $(u\circ k)(0)=u(k_{1}(0),0)$ and $(u\circ
k)(1)=u(k_{1}(1),1)=R(u(-k_{1}(0),0))$ are the same geodesic with opposite
orientation. Therefore these curves cannot both lie in the same component of
$W$, so we obtain a contradiction.

This proves that there must be two distinct simple geodesic loops at $p$ of
length at most $\max\\{l_{0},l_{1}/2\\}$ and $l_{1}$. Because $l_{0}\leq 8d$
and $l_{1}\leq 14d$, this proves our main theorem. ∎

#### Acknowledgements.

The author would like to thank Alexander Nabutovsky and Regina Rotman for
their invaluable guidance. This work was supported in part by the NSERC Canada
Graduate Scholarships Doctoral grant number 456419176. The author also
gratefully acknowledges support from the Institute for Advanced Study.

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