# Diastolic and isoperimetric inequalities on surfaces

Florent Balacheff  Florent Balacheff - Université des Sciences et
Technologies, Laboratoire Paul Painlevé, Bât. M2, 59655 Villeneuve d’Ascq,
France<EMAIL_ADDRESS>and  Stéphane Sabourau  Stéphane
Sabourau - Université François Rabelais, Tours. Laboratoire de Mathématiques
et Physique Théorique. CNRS, UMR 6083. Fédération de Recherche Denis Poisson
(FR 2964). Parc de Grandmont, 37400 Tours, France<EMAIL_ADDRESS>
###### Abstract.

We prove an universal inequality between the diastole, defined using a minimax
process on the one-cycle space, and the area of closed Riemannian surfaces.
Roughly speaking, we show that any closed Riemannian surface can be swept out
by a family of multi-loops whose lengths are bounded in terms of the area of
the surface. This diastolic inequality, which relies on an upper bound on
Cheeger’s constant, yields an effective process to find short closed geodesics
on the two-sphere, for instance. We deduce that every Riemannian surface can
be decomposed into two domains with the same area such that the length of
their boundary is bounded from above in terms of the area of the surface. We
also compare various Riemannian invariants on the two-sphere to underline the
special role played by the diastole.

Résumé. Nous démontrons une inégalité universelle entre la diastole, définie
par un procédé de minimax sur l’espace des $1$-cycles, et l’aire d’une surface
riemannienne fermée. De manière informelle, nous prouvons que toute surface
riemannienne fermée peut être balayée par une famille de multi-lacets dont les
longueurs sont contrôlées par l’aire de la surface. Cette inégalité
diastolique, qui repose sur une majoration de la constante de Cheeger, fournit
en particulier un procédé effectif pour trouver de courtes géodésiques fermées
sur une $2$-sphère. Nous déduisons que toute surface riemannienne peut être
décomposée en deux domaines de même aire dont la longueur du bord commun est
majorée à l’aide de l’aire de la surface. Nous comparons également divers
invariants riemanniens sur la $2$-sphère afin de souligner le rôle spécial
joué par la diastole.

###### Key words and phrases:

Cheeger constant, closed geodesics, curvature-free inequalities, diastole,
isoperimetric inequalities, one-cycles.

###### 2000 Mathematics Subject Classification:

Primary 53C23; Secondary 53C20, 58E10

The first author was supported by the Swiss National Science Foundation (grant
20-118014/1) during the redaction of this article. The second author has been
partially supported by the Swiss National Science Foundation.

##  1\. Introduction

The topology of a manifold $M$ and the topology of its loop space $\Lambda M$
are closely related, through their homotopy groups, for instance. The critical
points of the length or energy functionals on the loop space of a Riemannian
manifold can be studied using this connection. These critical points have a
special geometric meaning since they agree with the closed geodesics on the
manifold.

From the isomorphism between $\pi_{1}(\Lambda M,\Lambda^{0}M)$ and
$\pi_{2}(M)$ for the two-sphere (here, $\Lambda^{0}M$ denotes the space of
constant curves), G. D. Birkhoff proved the existence of a nontrivial closed
geodesic on every Riemannian two-sphere using a minimax argument on the loop
space (this method was extended in higher dimension by A. Fet and L.
Lyusternik, cf. [Kl78]).

Relationships on the lengths of these closed geodesics have then been
investigated. For instance, C. Croke [Cr88] showed that every Riemannian two-
sphere $M$ has a nontrivial closed geodesic of length

${\rm scg}(M)\leq 31\sqrt{{\rm area}(M)}.$ (1.1)

The notation ‘${\rm scg}$’ stands for the (length of a) shortest closed
geodesic. The inequality (1.1) has been improved in [NR02], [Sa04] and [Ro05].

The closed geodesic on the two-sphere obtained in C. Croke’s theorem does not
always arise from a minimax argument on the loop space, even though such an
argument is used in the proof. Indeed, the closed geodesics obtained from a
minimax argument on the loop space have a positive index when they are
nondegenerate, which is the case for generic (bumpy) metrics. Now, consider
two-spheres with constant area and three spikes arbitrarily long (cf. [Sa04,
Remark 4.10] for further detail). On these spheres, the closed geodesics
satisfying the inequality (1.1) have a null index while the closed geodesics
with positive index are as long as the spikes. Therefore, a minimax argument
on the loop space does not provide an effective way to bound the area of a
Riemannian two-sphere from below. More precisely, given a Riemannian two-
sphere $M$, we define the diastole, ${\rm dias}_{\Lambda}(M)$, over the loop
space $\Lambda M$ as

${\rm dias}_{\Lambda}(M):=\inf_{(\gamma_{t})}\sup_{0\leq t\leq 1}{\rm
length}(\gamma_{t})$

where $(\gamma_{t})$ runs over the families of loops inducing a generator of
$\pi_{1}(\Lambda M,\Lambda^{0}M)\simeq\pi_{2}(M)$. Then,

$\mbox{the ratio }\frac{{\rm dias}_{\Lambda}(M)}{\sqrt{{\rm area}(M)}}\mbox{
is unbounded}$ (1.2)

on the space of Riemannian metrics on $M$.

Note that a diastolic inequality between the diastole over the loop space and
the volume of the convex hypersurfaces in the Euclidean spaces does hold true,
cf. [Tr85], [Cr88].

The existence of a closed geodesic on the two-sphere can also be proved by
using a minimax argument on a different space, namely the one-cycle space
$\mathcal{Z}_{1}(M;{\mathbb{Z}})$. (Recall that one-cycles are unions of
loops, cf. Section 2 for a precise definition.) This minimax argument relies
on F. Almgren’s isomorphism [Al60] between
$\pi_{1}(\mathcal{Z}_{1}(M;{\mathbb{Z}}),\\{0\\})$ and
$H_{2}(M;{\mathbb{Z}})$, which holds true for every closed manifold $M$. Given
a closed Riemannian surface $M$, we define the diastole over the one-cycle
space as

${\rm dias}_{\mathcal{Z}}(M):=\inf_{(z_{t})}\sup_{0\leq t\leq
1}\mathbf{M}(z_{t})$ (1.3)

where $(z_{t})$ runs over the families of one-cycles inducing a generator of
$\pi_{1}(\mathcal{Z}_{1}(M;\Bbbk),\\{0\\})$ and $\mathbf{M}(z_{t})$ represents
the mass (or length) of $z_{t}$. Here $\Bbbk={\mathbb{Z}}$ if $M$ is
orientable and $\Bbbk={\mathbb{Z}}/2{\mathbb{Z}}$ otherwise. For short, we
will write ${\rm dias}(M)$ for ${\rm dias}_{\mathcal{Z}}(M)$. From a result of
J. Pitts [Pi74, p. 468], [Pi81, Theorem 4.10] (see also [CC92]), this minimax
principle gives rise to a union of closed geodesics (counted with
multiplicity) of total length ${\rm dias}(M)$. Hence, ${\rm scg}(M)\leq{\rm
dias}(M)$. This principle has been used in [CC92], [NR02], [Sa04], [Ro05],
[Ro06], [Ba08] and [Sa09] in the study of closed geodesics on Riemannian two-
spheres.

On nonsimply connected surfaces, no minimax principle is required to show the
existence of a closed geodesic. In this setting, we directly define the
systole as the minimum of the length functional over the connected components
not containing the trivial loops of the loop space. Every closed Riemannian
surfaces $M$ of genus $g\geq 1$ satisfies the following asymptotically optimal
systolic inequality

${\rm sys}(M)\leq C\,\frac{\log(g+1)}{\sqrt{g}}\,\sqrt{{\rm area}(M)}$ (1.4)

where ${\rm sys}(M)$ is the systole of $M$ and $C$ is a universal constant,
cf. [Gr83], [Ba04] and [KS05] for three different proofs.

The goal of this article is to establish curvature-free inequalities similar
to (1.1) and (1.4). The use of the one-cycle space, rather than the loop
space, introduces some flexibility. It allows us to cut and paste closed
curves, and to deal with both simply and nonsimply connected surfaces. We show
a difference of nature between the diastole over the loop space and the
diastole over the one-cycle space on the two-sphere, and between the systole
and the diastole for surfaces of large genus. More precisely, we obtain the
following diastolic inequality.

######  Theorem 1.1.

There exists a positive constant $C\leq 10^{8}$ such that every closed
Riemannian surface $M$ of genus $g\geq 0$ satisfies

${\rm dias}(M)\leq C\,\sqrt{g+1}\sqrt{{\rm area}(M)}.$ (1.5)

Since the minimax principle (1.3) gives rise to a union of closed geodesics of
length ${\rm dias}(M)$, Theorem 1.1 yields a construction of short closed
geodesics on surfaces through Morse theory over the one-cycle space. In
particular, it sheds some light on the nature of the closed geodesics whose
lengths provide a lower bound on the area of the two-spheres. Compare with
(1.1) and (1.2).

The dependence on the genus in the inequality (1.5) is optimal, cf. Remark
7.3, and should be compared with the one in (1.4).

A version of the diastolic inequality (1.5) holds true for compact surfaces
with boundary, in particular for disks, cf. Remark 2.4.

The inequality (1.5) is derived from a stronger estimate, where the diastole
is replaced with the technical diastole introduced in Definition 2.2. This
estimate also yields the following result.

######  Corollary 1.2.

There exists a positive constant $C$ such that every closed Riemannian surface
$M$ of genus $g\geq 0$ decomposes into two domains with the same area whose
length of their common boundary $\gamma$ satisfies

${\rm length}(\gamma)\leq C\,\sqrt{g+1}\sqrt{{\rm area}(M)}.$ (1.6)

The two domains in the previous result are not necessarily connected, even on
two-spheres. A counterexample is given by two-spheres with three long fingers,
whose area is equally concentrated in the tips of these fingers (cf. [Sa04,
Remark 4.10]).

The second author [Sa04] showed that the area of a bumpy Riemannian two-sphere
can be bounded from below in terms of the length of its shortest one-cycle of
index one. Our inequality (1.5) on the two-sphere extends this result since
the minimax process used in the definition of the diastole gives rise to a
one-cycle of index one when the metric is bumpy. However, the relation between
the filling radius of a bumpy Riemannian two-sphere and the length of its
shortest one-cycle of index one established in [Sa04] cannot be extended to
the diastole. Indeed, from [Sa04, Theorem 1.6], there exists a sequence
$M_{n}$ of Riemannian two-spheres such that

$\lim_{n\to+\infty}\frac{{\rm FillRad}(M_{n})}{{\rm dias}(M_{n})}=0.$

This result illustrates the difference of nature between the length of the
shortest closed geodesic or of the shortest one-cycle of index one, which can
be bounded by the filling radius on the two-sphere, and the diastole. It also
shows that the proof of Theorem 1.1 requires different techniques.

The arguments used throughout this article are rather “elementary” and robust.
It is our hope that they can be adapted to investigate further problems.

At the end of this article, we consider a minimax principle on the one-cycle
space for another functional and establish further geometric inequalities on
the two-sphere.

Higher dimensional analogs of the diastole, sometimes called $k$-widths, have
been investigated in [Al65], [Pi81], [Gr83] and [Gu07]. In particular, L. Guth
[Gu07] obtained upper bounds on the $k$-width of Euclidean domains in terms of
their $n$-dimensional volumes. He also showed that no such bound holds in the
Riemannian setting for the $(n-1)$-width with $n\geq 3$. Our main result
provides a positive result on Riemannian surfaces, answering a question raised
in [Gu07, p. 1148] in a particular case.

Let us present the structure of the article and an outline of the proof of the
diastolic inequality (1.5). The one-cycle space and the definition of the
diastole are presented in Section 2. In Section 3, we show how to replace a
Riemannian surface with a simplicial piecewise flat surface with comparable
area and diastole. This will enable us to prove the main diastolic inequality
by induction on the number of simplices in an approaching piecewise flat
surface. In Section 4, we prove an upper bound on Cheeger’s constant, and on
some related invariant, in terms of the area of the surface alone. This upper
bound yields an isoperimetric inequality which permits us to split the surface
into two domains $D_{1}$ and $D_{2}$ whose boundary lengths are small in
comparison to their areas. This isoperimetric inequality can be thought of as
a discrete version of the diastolic inequality. At this stage, we could cap
off the two domains $D_{1}$ and $D_{2}$ and apply the discrete version of the
diastolic inequality to the two resulting surfaces with the hope to derive a
continuous version of it by iterating this principle sufficiently many times.
Unfortunately, passing from a discrete parameter family of one-cycles to a
continuous parameter family on a smooth surface is technically challenging.
Arguing by induction on the number of simplices of a piecewise flat surface
approaching the initial surface turns out to be more manageable in this case.
More specifically, we argue as follows. In Section 5, we prove a simplicial
version of the isoperimetric inequality obtained from the upper bound on
Cheeger’s constant. The two simplicial surfaces obtained by coning off the
domains $D_{1}$ and $D_{2}$ have fewer simplices than the initial simplicial
surface. Using a cut-and-paste argument on the families of one-cycles, we
compare the diastole of the two new simplicial surfaces to the diastole of the
initial simplicial surface in Section 6. The diastolic inequality is then
established by induction on the number of simplices and the genus in Section
7. Further related geometric inequalities about families of one-cycles
sweeping out a Riemannian two-sphere are presented in Section 8 and in the
appendix.

##  2\. Almgren’s isomorphism and definition of the diastole

We introduce the space of flat chains and cycles with coefficients in
${\mathbb{Z}}$ or ${\mathbb{Z}}/2{\mathbb{Z}}$, defined in [Fl66] and [Fe69].

Let $M$ be a closed Riemannian manifold, $N$ be a submanifold of $M$ (possibly
with boundary), $\Bbbk={\mathbb{Z}}$ if $M$ is orientable and
$\Bbbk={\mathbb{Z}}/2{\mathbb{Z}}$ otherwise, and $k$ be an integer.

A Lipschitz $k$-chain of $M$ with coefficients in $\Bbbk$ is a finite sum
$C=\sum a_{i}f_{i}$ where $a_{i}\in\Bbbk$ and $f_{i}$ is a Lipschitz map from
the standard $k$-simplex to $M$. The volume of $C$, denoted by $|C|$, is
defined as $\sum|a_{i}|{\rm vol}(f_{i}^{*}{\mathcal{G}})$ where
${\mathcal{G}}$ is the Riemannian metric on $M$. The flat pseudo-norm of a
Lipschitz $k$-chain $C$ is the infimum of $|C-\partial D|+|D|$ over the
Lipschitz $(k+1)$-chains $D$. Two Lipschitz chains $C_{1}$ and $C_{2}$ are
equivalent if the flat pseudo-norm between them is zero. The flat pseudo-norm
induces a distance (flat norm) on the space of equivalent classes of Lipschitz
$k$-chains. By definition, the completion of this metric space, denoted by
$C_{k}(M;\Bbbk)$, is the space of (flat) $k$-chains with coefficients in
$\Bbbk$.

The mass functional on $C_{k}(M;\Bbbk)$, denoted by $\mathbf{M}$, is defined
as the largest lower-semicontinuous extension of the volume functional on the
space of Lipschitz $k$-chains. The flat norm of a $k$-chain can be computed by
using the definition of the flat pseudo-norm with $D$ running over the
$(k+1)$-chains and the mass $\mathbf{M}$ replacing the volume functional.

The space of $k$-chains of $M$ relative to $N$ with coefficients in $\Bbbk$ is
defined as the quotient $C_{k}(M,N;\Bbbk):=C_{k}(M;\Bbbk)/C_{k}(N;\Bbbk)$
endowed with the quotient flat norm and the quotient mass still denoted
$\mathbf{M}$. The boundary map of a Lipschitz $k$-chain induces a boundary map
$\partial:C_{k+1}(M,N;\Bbbk)\to C_{k}(M,N;\Bbbk)$ for every $k\in{\mathbb{N}}$
such that $\partial^{2}=0$ which gives rise to a complex

$\ldots\rightarrow
C_{k+1}(M,N;\Bbbk)\overset{\partial}{\rightarrow}C_{k}(M,N;\Bbbk)\rightarrow\ldots$

The cycles of this complex are by definition the $k$-cycles of $M$ relative to
$N$ with coefficients in $\Bbbk$. They can be represented by $k$-chains of $M$
with boundary lying in $N$. We denote by $\mathcal{Z}_{k}(M,N;\Bbbk)$ the
space formed of these $k$-cycles. When $N$ is empty, we simply write
$\mathcal{Z}_{k}(M;\Bbbk)$ or $\mathcal{Z}_{k}(M)$.

We describe now the structure of the zero- and one-cycles. A zero-cycle is
indecomposable if it induced by a point $p$, in which case, its mass is equal
to $1$. Every zero-cycle $z$ decomposes into a finite sum of indecomposable
zero-cycles $p_{i}$, i.e., $z=\sum_{i=1}^{n}p_{i}$, such that
$\mathbf{M}(z)=n$. The structure of the one-cycles is the following. A one-
cycle is indecomposable if it is induced by a simple closed curve or a simple
arc with endpoints in $N$. Every one-cycle $z$ decomposes (not necessarily in
a unique way) into a sum of indecomposable one-cycles $z_{i}$, i.e.,
$z=\sum_{i\in{\mathbb{N}}}z_{i}$, such that
$\mathbf{M}(z)=\sum_{i\in{\mathbb{N}}}{\rm length}(z_{i})$ (see [Fe69, p.
420]).

The homotopy groups of the space of (relative) cycles have been determined by
F. Almgren, cf. [Al60], [Al65, §13.4] and [Pi81, §4.6]. On surfaces, one has
the following natural isomorphism for the one-cycle space

$\pi_{1}(\mathcal{Z}_{1}(M;\Bbbk),\\{0\\})\simeq H_{2}(M;\Bbbk)\simeq\Bbbk.$
(2.1)

More generally, if $N$ is a submanifold with boundary of $M$, one has

$\pi_{1}(\mathcal{Z}_{1}(M,N;\Bbbk),\\{0\\})\simeq H_{2}(M,N;\Bbbk).$ (2.2)

This isomorphism permits us to apply the Almgren-Pitts minimax principle to
the one-cycle space of surfaces. Recall that $\Bbbk={\mathbb{Z}}$ if the
surface is orientable and that $\Bbbk={\mathbb{Z}}/2{\mathbb{Z}}$ otherwise.

Let us consider the one-parameter families $(z_{t})_{0\leq t\leq 1}$ of one-
cycles sweeping out a closed surface $M$, that is, which satisfy the following
conditions

  1. (D.1)

$z_{t}$ starts and ends at the null one-cycle;

  2. (D.2)

$z_{t}$ induces a generator of
$\pi_{1}(\mathcal{Z}_{1}(M;\Bbbk),\\{0\\})\simeq\Bbbk$.

For technical reasons, we will also consider the following conditions

  1. (D.3)

there exists a finite subdivision $t_{1}<t_{2}<\cdots<t_{k}$ of $[0,1]$ and
homotopies $(\gamma_{j,t})_{t_{p}\leq t\leq t_{p+1}}$, $j\in J_{p}$, of
finitely many piecewise smooth loops such that the one-cycle $z_{t}$ agrees
with the finite sum $\sum_{j\in J_{p}}\gamma_{j,t}$ on $[t_{p},t_{p+1}]$;

  2. (D.4)

for every $t\in]t_{p},t_{p+1}[$, the loops $\gamma_{j,t}$, $j\in J_{p}$, are
simple and disjoint;

  3. (D.5)

the loops $\gamma_{j,t}$ and $\gamma_{j^{\prime},t^{\prime}}$ are disjoint for
$t\neq t^{\prime}$.

######  Example 2.1.

Let $f:M\to{\mathbb{R}}$ be a Morse function on a closed surface $M$. The
level curves $z_{t}=f^{-1}(t)$ define a one-parameter family of one-cycles
$(z_{t})$ satisfying the conditions (D.1-5).

######  Definition 2.2.

The diastole of $M$, denoted by ${\rm dias}(M)$, is defined as the minimax
value

${\rm dias}(M):=\inf_{(z_{t})}\sup_{0\leq t\leq 1}\mathbf{M}(z_{t})$ (2.3)

where $(z_{t})$ runs over the families of one-cycles satisfying the conditions
(D.1-2) above. Similarly, the (technical) diastole of $M$, denoted by ${\rm
dias}^{\prime}(M)$, is defined as the minimax value

${\rm dias}^{\prime}(M):=\inf_{(z_{t})}\sup_{0\leq t\leq 1}\mathbf{M}(z_{t})$
(2.4)

where $(z_{t})$ runs over the families of one-cycles satisfying the conditions
(D.1-5) above.

######  Remark 2.3.

We will prove that the diastolic inequality (1.5) holds true if one replaces
the diastole ${\rm dias}(M)$ with the (technical) diastole ${\rm
dias}^{\prime}(M)$. Since ${\rm dias}(M)\leq{\rm dias}^{\prime}(M)$, this will
yield Theorem 1.1.

######  Remark 2.4.

If $M$ is a Riemannian compact surface with boundary $\partial M$, the
isomorphism (2.2) allows us to define a relative notion of diastole using one-
cycles relative to $\partial M$. This relative diastole on $M$ is less or
equal to the diastole on the closed Riemannian surface obtained by filling the
boundary components of $M$ with disks of small area. Therefore, a diastolic
inequality similar to (1.5) holds for compact surfaces with boundary where the
diastole is replaced with the diastole relative to the boundary.

##  3\. From Riemannian metrics to simplicial surfaces

######  Proposition 3.1.

Every closed Riemannian surface $M$ is $K$-bilipschitz homeomorphic to a
closed piecewise flat surface $M_{0}$ triangulated by equilateral flat
triangles, where $K=33$.

######  Remark 3.2.

If we do not require the triangles of $M_{0}$ to have the same size, we can
take the constant $K$ to be arbitrarily close to one in the previous
proposition. However, this is no longer true for surfaces with nonzero Euler
caracteristic when all the triangles of $M_{0}$ are supposed to be
equilateral.

Let us argue by contradiction. Subdividing the triangulation of $M_{0}$ if
necessary, we can assume that the equilateral flat triangles of $M_{0}$ are
arbitrarily small. Consider the geodesic triangulation $\mathcal{T}$ of $M$
isotopic to the image of the triangulation of $M_{0}$ under the
$K$-bilipschitz homeomorphism while keeping the vertices fixed. If $K$ is
close enough to one, the angles of all the triangles of $\mathcal{T}$ are
close to $\frac{\pi}{3}$.

From Euler’s formula, we have the following relation

$6\chi(M)=\sum_{i}(6-v_{i})$

between the Euler characteristic of $M$ and the degrees $v_{i}$ of the
vertices of its triangulation. If the Euler characteristic of $M$ is nonzero,
at least one vertex of the triangulation of $M$ has its degree different from
$6$. In this case, the angles of the triangulation $\mathcal{T}$ of the smooth
surface $M$ cannot all be close to $\frac{\pi}{3}$. Hence a contradiction.

For the two-torus and the Klein bottle, the constant $K$ in the previous
proposition can be taken to be arbitrarily close to one, see [CVM90] and the
following.

The proof of Proposition 3.1 rests on the following construction.

Let $M$ be a closed Riemannian surface. Choose
$\varepsilon\in]0,\frac{1}{100}[$ and
$\varepsilon^{\prime}\in]0,\varepsilon[$. From [CVM90], there exists a
geodesic triangulation $\mathcal{T}$ of $M$ such that

  1. (1)

the diameter of every triangle of $\mathcal{T}$ is less than
$\varepsilon^{\prime}$;

  2. (2)

the angles of every triangle of $\mathcal{T}$ are between
$\frac{2\pi}{7}-\varepsilon^{\prime}$ and
$\frac{5\pi}{14}+\varepsilon^{\prime}$.

Denote by $\Delta_{i}$ the triangles of the triangulation $\mathcal{T}$. By
taking $\varepsilon^{\prime}$ small enough, we can assume that every triangle
$\Delta_{i}$ of $\mathcal{T}$ is almost isometric to a triangle
$\Delta_{i}^{\prime}$ of the Euclidean plane with the same side lengths as
$\Delta_{i}$. More precisely, we can assume that

  1. (a)

there exists a $(1+\varepsilon)$-bilipschitz homeomorphism between
$\Delta_{i}$ and $\Delta_{i}^{\prime}$;

  2. (b)

the diameter of the triangle $\Delta_{i}^{\prime}$ is less than $\varepsilon$;

  3. (c)

the angles of the triangle $\Delta_{i}^{\prime}$ are strictly between
$\frac{\pi}{4}$ and $\frac{3\pi}{7}$.

Replacing the triangles $\Delta_{i}$ of the triangulation $\mathcal{T}$ of $M$
by the Euclidean triangles $\Delta_{i}^{\prime}$ gives rise to a piecewise
flat metric with conical singularities on the surface. The size of the
Euclidean triangles making up this piecewise flat metric may vary from one
triangle to another. By subdividing some of these triangles, one can make this
size more uniform.

More precisely, the segments connecting the midpoints of the sides of each
Euclidean triangle $\Delta_{i}^{\prime}$ decompose $\Delta_{i}^{\prime}$ into
four triangles twice smaller than $\Delta_{i}^{\prime}$ with the same angles
as $\Delta_{i}^{\prime}$ (see Figure 1).

$\Delta^{\prime\prime}_{i,1}$ $\Delta^{\prime\prime}_{i,2}$
$\Delta^{\prime\prime}_{i,3}$ $\Delta^{\prime\prime}_{i,4}$

Figure 1. Decomposition of a triangle $\Delta^{\prime}_{i}$

By iterating this subdivision process, replacing some of the
$\Delta_{i}^{\prime}$’s by four triangles twice smaller than
$\Delta_{i}^{\prime}$, one can construct a new collection
$\mathcal{T}^{\prime\prime}$ of triangles $\Delta_{j}^{\prime\prime}$ such
that

$\max_{j}{\rm diam}(\Delta_{j}^{\prime\prime})\leq 2\,\min_{j}{\rm
diam}(\Delta_{j}^{\prime\prime}).$

######  Lemma 3.3.

There exists $\ell>0$ such that every triangle of $\mathcal{T}^{\prime\prime}$
is $8$-bilipschitz homeomorphic to the equilateral flat triangle of side
length $\ell$ through an affine map.

###### Proof.

Let $\ell=\max_{j}{\rm diam}(\Delta_{j}^{\prime\prime})$ and
$\Delta_{j}^{\prime\prime}$ be a triangle of $\mathcal{T}^{\prime\prime}$. By
construction, the angles of $\Delta_{j}^{\prime\prime}$ strictly lie between
$\frac{\pi}{4}$ and $\frac{3\pi}{7}$ and the length of its longest side
between $\frac{\ell}{2}$ and $\ell$.

Consider the longest side of $\Delta_{j}^{\prime\prime}$. Let $\delta$ be the
perpendicular bisector of this side and $v$ be the vertex of
$\Delta_{j}^{\prime\prime}$ opposite to this side. Move $v$ to its orthogonal
projection on $\delta$. Then, move this projection along the perpendicular
bisector $\delta$ so as to obtain an equilateral flat triangle. A
straightforward computation shows that the composite of these two
transformations defines a $4$-bilipschitz affine homeomorphism from
$\Delta_{j}^{\prime\prime}$ onto an equilateral flat triangle (recall that the
angles of $\Delta_{j}^{\prime\prime}$ are greater than $\frac{\pi}{4}$).

The length of the sides of this equilateral triangle is between
$\frac{\ell}{2}$ and $\ell$. Therefore, a homothety of ratio between $1$ and
$2$ transforms this equilateral triangle into an equilateral flat triangle of
side length $\ell$. ∎

The collection $\mathcal{T}^{\prime\prime}$ of these triangles may not form a
triangulation of $M$ since the vertex of a triangle
$\Delta_{j}^{\prime\prime}$ may lie in the interior of the edge of another
triangle. However, the union of the triangles of $\mathcal{T}^{\prime\prime}$
covers $M$ and the interiors of two distinct triangles of
$\mathcal{T}^{\prime\prime}$ are disjoint. Furthermore, if $e_{1}$ and $e_{2}$
are the edges of two triangles of $\mathcal{T}^{\prime\prime}$ then either
$e_{1}$ lies in $e_{2}$ or $e_{2}$ lies in $e_{1}$, unless the two edges are
disjoint or intersect at a single point.

Let us see now how to obtain a genuine triangulation of $M$. For that purpose,
we will need the following lemma.

######  Lemma 3.4.

Consider an edge of a triangle of $\mathcal{T}^{\prime\prime}$ which contains
at least two edges of some other triangles of $\mathcal{T}^{\prime\prime}$.
Then this edge agrees exactly with the union of these two edges.

###### Proof.

Since the angles of every triangle $\Delta^{\prime\prime}$ of
$\mathcal{T}^{\prime\prime}$ are greater than $\frac{\pi}{4}$, the length of
the longest side of $\Delta^{\prime\prime}$, which agrees with its diameter,
is less than twice the length of its shortest side. This remark is left to the
reader.

Consider an edge $e$ of a triangle $\Delta^{\prime\prime}$ of
$\mathcal{T}^{\prime\prime}$ which contains at least two edges of some other
triangles of $\mathcal{T}^{\prime\prime}$. By construction, the edge $e$ is
made up of a necessarily even number of edges of some other triangles of
$\mathcal{T}^{\prime\prime}$. Assume that at least four edges of some other
triangles of $\mathcal{T}^{\prime\prime}$ lie in $e$. The length of these
edges is at most one quarter of the length of $e$. Therefore, our initial
remark shows that there exist some triangles of $\mathcal{T}^{\prime\prime}$
whose diameter is less than half the diameter of $\Delta^{\prime\prime}$.
Hence a contradiction with the construction of $\mathcal{T}^{\prime\prime}$. ∎

If the edge $e$ of a triangle $\Delta^{\prime\prime}$ of
$\mathcal{T}^{\prime\prime}$ agrees with the union of the edges of two other
triangles of $\mathcal{T}^{\prime\prime}$, then we split
$\Delta^{\prime\prime}$ along the segment connecting the midpoint of $e$ to
the vertex of $\Delta^{\prime\prime}$ opposite to that edge. The new
collection of triangles still satisfies the conclusion of the previous lemma.
We repeat this process to this new collection until we get a genuine
triangulation $\mathcal{T}_{0}$ of $M$, where every edge is the side of
exactly two adjacent triangles.

######  Lemma 3.5.

Every triangle of $\mathcal{T}_{0}$ is $32$-bilipschitz homeomorphic to the
equilateral flat triangle of side length $\ell$ (the same $\ell$ as in Lemma
3.3).

###### Proof.

The ($8$-bilipschitz) affine homeomorphism of Lemma 3.3 takes segments to
segments and the midpoints of segments to the midpoints of the image segments.
Therefore, we can work with an equilateral flat triangle $\Delta$ of side
length $\ell$ instead of a triangle of $\mathcal{T}^{\prime\prime}$. From
Lemma 3.4, this triangle is split into at most four triangles through the
subdivision of $\mathcal{T}^{\prime\prime}$ into $\mathcal{T}_{0}$. These
subtriangles of $\Delta$ are isometric to one of the following

  1. (1)

the equilateral triangle of side length $\frac{\ell}{2}$;

  2. (2)

the isosceles triangle with a principal angle of $\frac{2\pi}{3}$ and two
sides of length $\frac{\ell}{2}$;

  3. (3)

the right triangle with angles $\frac{\pi}{2}$, $\frac{\pi}{3}$ and
$\frac{\pi}{6}$, and side lengths $\ell$, $\frac{\sqrt{3}}{2}\ell$ and
$\frac{\ell}{2}$.

Since each of these triangles is $4$-bilipschitz homeomorphic to $\Delta$, we
obtain the desired result. ∎

We can now conclude.

###### Proof of Proposition 3.1.

Denote by $M_{0}$ the piecewise flat surface obtained by replacing every
triangle of $\mathcal{T}_{0}$ by the equilateral flat triangle with side
length $\ell$. Putting together the bilipschitz homeomorphisms of (a) above
and Lemma 3.5 gives rise to a $(1+\varepsilon)32$-bilipschitz homeomorphism
between the Riemannian surface $M$ and the piecewise flat surface $M_{0}$. ∎

######  Corollary 3.6.

The area, the diameter and the diastole of $M$ and $M_{0}$ satisfy the
following inequalities

$\displaystyle K^{-2}$ $\displaystyle\leq\frac{{\rm area}(M_{0})}{{\rm
area}(M)}\leq K^{2}$ (3.1) $\displaystyle K^{-1}$ $\displaystyle\leq\frac{{\rm
dias}(M_{0})}{{\rm dias}(M)}\leq K$ (3.2)

where $K=33$.

######  Remark 3.7.

From Corollary 3.6, proving a diastolic inequality on $M$ amounts to proving a
diastolic inequality on the simplicial surface $M_{0}$. Since diastolic
inequalities are scale invariant, the size of the equilateral flat triangles
composing the piecewise flat surface $M_{0}$ can arbitrarily be fixed equal to
$1$. We will always assume this is the case when we consider simplicial
surfaces.

##  4\. Upper bounds on Cheeger’s constant

Let $M$ be a closed Riemannian surface. Recall that the Cheeger constant
$h(M)$ of $M$ is defined as

$h(M)=\inf_{D}\frac{{\rm length}(\partial D)}{\min\\{{\rm area}(D),{\rm
area}(M\setminus D)\\}},$

where the infimum is taken over all the domains $D$ of $M$ with smooth
boundary.

######  Proposition 4.1.

Every closed Riemannian surface $M$ (not necessarily orientable) of genus
$g\geq 0$ satisfies the following inequality

$h(M)\leq\frac{\sqrt{96\pi(g+1)}}{\sqrt{{\rm area}(M)}}.$ (4.1)

###### Proof.

Let $\lambda_{1}(M)$ be the first nonzero eigenvalue of the Laplacian on $M$.
From [Ch70], we have

$\lambda_{1}(M)\geq\frac{h(M)^{2}}{4}.$

On the other hand, from [LY82], we have

$\lambda_{1}(M)\,{\rm area}(M)\leq 24\pi(g+1).$ (4.2)

The combination of these two inequalities yields the desired upper bound on
the Cheeger constant. ∎

######  Remark 4.2.

For orientable surfaces, the inequality (4.2) still holds by replacing the
constant $24$ by $8$, cf. [YY80]. This leads to a better upper bound on
Cheeger’s constant in Proposition 4.1, where the constant $96$ can be replaced
by $32$. Thus, Proposition 4.1 improves the multiplicative constant of an
inequality established in [Pa09] by using a different method.

Note that the dependence of the upper bound on the genus cannot be
significantly improved. Indeed, there exist closed hyperbolic surfaces of
arbitrarily large genus with Cheeger constant bounded away from zero, cf.
[Br86] for instance.

An upper bound on the Cheeger constant of Riemannian two-spheres similar to
(4.1) can be obtained using different arguments. Although these arguments do
not lead to a better multiplicative constant, they rely on a comparison
between two intermediate invariants, namely the filling radius and the
invariant ${\mathcal{L}}$, cf. below, which is interesting in its own right.
Further, they can be generalized in higher dimension (but the final statement
is not as meaningful). For these reasons, we still include this result even
though we will not use it in the rest of this paper.

Let $M$ be a Riemannian two-sphere. Define

${\mathcal{L}}(M)=\inf_{\gamma}{\rm length}(\gamma)$

where $\gamma$ runs over all the simple loops dividing $M$ into two disks of
area at least $\frac{1}{4}{\rm area}(M)$. Clearly,

$h(M)\leq 4\,\frac{{\mathcal{L}}(M)}{{\rm area}(M)}.$

We show in Proposition 4.4 that the invariant ${\mathcal{L}}$ provides a lower
bound on the filling radius on every Riemannian two-sphere.

######  Definition 4.3.

Let $i:M\hookrightarrow L^{\infty}(M)$ be the Kuratowski distance preserving
embedding defined by $i(x)(.)=d_{M}(x,.)$. The _filling radius_ of $M$,
denoted by ${\rm FillRad}(M)$, is defined as the infimum of the positive reals
$r$ such that the homomorphism $H_{n}(M;{\mathbb{Z}})\longrightarrow
H_{n}(U_{r};{\mathbb{Z}})$ induced by the embedding of $M$ into the
$r$-tubular neighborhood $U_{r}$ of its image $i(M)$ in $L^{\infty}(M)$ is
trivial.

Recall that a lower bound on the filling radius provides bounds on the area
and the diameter of Riemannian two-spheres since ${\rm
FillRad}(M)\leq\sqrt{{\rm area}(M)}$ from [Gr83] and ${\rm
FillRad}(M)\leq\frac{1}{3}{\rm diam}(M)$ from [Ka83].

######  Proposition 4.4.

Let $M$ be a Riemannian two-sphere. Then,

${\mathcal{L}}(M)\leq 6\,{\rm FillRad}(M).$

In particular,

$h(M)\leq\frac{24}{\sqrt{{\rm area}(M)}}.$

######  Remark 4.5.

Similar lower bounds on the filling radius have been proved in [Gr83] for
essential manifolds, where ${\mathcal{L}}$ is replaced with the systole, and
in [Sa04] on the two-sphere, where ${\mathcal{L}}$ is replaced with the
shortest length of a geodesic loop.

###### Proof.

We argue by contradiction. Pick two reals $r$ and $\varepsilon$ such that

${\rm FillRad}(M)<r<r+\varepsilon<\frac{1}{6}{\mathcal{L}}(M).$

Let $i:M\hookrightarrow L^{\infty}(M)$ be the Kuratowski distance preserving
embedding defined by $i(x)(.)=d_{M}(x,.)$. By definition of the filling
radius, there exists a map $\sigma:P\to U_{r}$ from a $3$-complex $P$ to the
$r$-tubular neighborhood $U_{r}$ of $i(M)$ in $L^{\infty}=L^{\infty}(M)$ such
that $\sigma_{|\partial P}:\partial P\to i(M)$ represents the fundamental
class of $i(M)\simeq M$ in $H_{2}(i(M);{\mathbb{Z}})\simeq
H_{2}(M;{\mathbb{Z}})$.

Let us show that the map $\sigma_{|\partial P}:\partial P\to i(M)$ extends to
a map $f:P\to i(M)$. This would yield the desired contradiction since the
fundamental class of $i(M)$, represented by $\sigma_{|\partial P}$, is
nonzero.

Deforming $\sigma$ and subdividing $P$ if necessary, we can assume that
$\sigma$ takes every edge of $P$ to a minimizing segment and that the diameter
of the $\sigma$-image of every simplex in $P$ is less than $\varepsilon$. We
define $f$ to agree with $\sigma$ on the $2$-complex $\partial P$.

We now define $f$ on the $0$-skeleton of $P\setminus\partial P$ by sending
each vertex $p$ of $P\setminus\partial P$ to a point $f(p)$ on $i(M)$ closest
to $\sigma(p)$. Thus, $d_{L^{\infty}}(f(p),\sigma(p))<r$ for every vertex $p$
of $P$. Since the embedding $i$ preserves the distances, every pair of
adjacent vertices $p,q\in P$ satisfies

$\displaystyle d_{i(M)}(f(p),f(q))$ $\displaystyle=$ $\displaystyle
d_{L^{\infty}}(f(p),f(q))$ $\displaystyle\leq$ $\displaystyle
d_{L^{\infty}}(f(p),\sigma(p))+d_{L^{\infty}}(\sigma(p),\sigma(q))+d_{L^{\infty}}(\sigma(q),f(q))$
$\displaystyle<$ $\displaystyle 2r+\varepsilon=:\rho$

We extend $f$ to the $1$-skeleton of $P$ by mapping each edge of
$P\setminus\partial P$ to a minimizing segment joining the images of the
endpoints. By construction, the image under $f$ of the boundary of every
$2$-simplex $\Delta^{2}$ of $P$ is a simple loop $\gamma_{\Delta^{2}}$ of
length less than $3\rho$. Since $3\rho<{\mathcal{L}}(M)$ for $\varepsilon$
small enough, the simple loop $\gamma_{\Delta^{2}}$ bounds a disk
$D_{\Delta^{2}}$ in $i(M)\simeq M$ of area less than $\frac{1}{4}{\rm
area}(M)$. We extend $f$ to the $2$-skeleton of $P$ by mapping each
$2$-simplex $\Delta^{2}$ to the disk $D_{\Delta^{2}}$. Thus, the restriction
$\partial\Delta^{3}\to i(M)$ of $f$ to the boundary $\partial\Delta^{3}$ of a
$3$-simplex $\Delta^{3}$ of $P$ is a degree zero map between two spheres since
the area of its image is less than the area of $i(M)\simeq M$. Therefore, $f$
extends to each simplex of $P$ and gives rise to an extension $f:P\to i(M)$ of
$\sigma_{|\partial P}$. ∎

##  5\. Isoperimetric inequalities on simplicial surfaces

We will need the following remark. Let $\varepsilon>0$. Every piecewise flat
metric with conical singularities on a surface can be smoothed out at its
singularities into a Riemannian metric $(1+\varepsilon)$-bilipschitz
homeomorphic to it. Therefore, Proposition 4.1 still holds true for closed
simplicial surfaces.

We prove now that the proposition 4.1 still holds in a simplicial setting.

######  Proposition 5.1.

Let $M_{0}$ be a closed simplicial surface (not necessarily orientable) of
genus $g\geq 0$ formed of $N$ triangles. The surface $M_{0}$ decomposes into
two simplicial domains, $D_{1}$ and $D_{2}$, with disjoint interiors,
satisfying the following inequality for $i=1,2$

${\rm length}(\partial D_{i})\leq C_{0}\sqrt{g+1}\,\frac{\min\\{{\rm
area}(D_{1}),{\rm area}(D_{2})\\}}{\sqrt{{\rm area}(M_{0})}},$ (5.1)

where $C_{0}=15\sqrt{96\pi}$.

###### Proof.

Assume that $N\geq N_{*}$ where $N_{*}:=10\cdot 96\pi(g+1)$. Fix
$\varepsilon\in]0,\frac{1}{50}[$. Since $M_{0}$ is composed of $N$ equilateral
flat triangles (of side length $1$), the area of $M_{0}$ is equal to

${\rm area}(M_{0})=\frac{\sqrt{3}}{4}N.$ (5.2)

From the remark at the beginning of this section, the surface $M_{0}$
decomposes into two (not necessarily simplicial) domains $D_{1}$ and $D_{2}$
with common boundary $\delta$ such that

${\rm
length}(\delta)<(1+\varepsilon)\,\sqrt{96\pi}\,\sqrt{g+1}\,\frac{\min\\{{\rm
area}(D_{1}),{\rm area}(D_{2})\\}}{\sqrt{{\rm area}(M_{0})}}.$ (5.3)

Combined with the formula (5.2) and the bound $N\geq N_{*}$, this estimate
yields

${\rm length}(\delta)<\frac{1}{2}\,{\rm area}(D_{j})\text{ for }j=1,2.$ (5.4)

We want to deform $\delta$ into a one-cycle lying in the one-skeleton of
$M_{0}$ while controlling its length and the area of the two domains it
bounds.

Without loss of generality, we can suppose that

  1. (1)

$\delta$ is composed of finitely many simple loops;

  2. (2)

each of these loops meets the edges of the triangulation of $M_{0}$;

  3. (3)

each of these loops intersects the edges of the triangulation at finitely many
points;

  4. (4)

none of these loops passes through the midpoints of the edges of the
triangulation.

In this situation, the one-cycle $\delta$ decomposes into a finite union of
subarcs $\tau_{i}$, where $i\in I$, such that each arc $\tau_{i}$ is contained
into a triangle $\Delta$ of the triangulation of $M_{0}$ with its endpoints
lying in the boundary $\partial\Delta$ of $\Delta$. For every $i\in I$, denote
by $\ell_{i}$ the length of $\tau_{i}$. We have

${\rm length}(\delta)=\sum_{i\in I}\ell_{i}.$

For every triangle $\Delta$ of the triangulation of $M_{0}$, we apply the
curve-shortening process defined in [Sa04, §2.2] to the finite collection of
arcs $\tau_{i}$ lying in the convex domain $\Delta$. Through this curve-
shortening process, the arcs $\tau_{i}$ remain disjoint, do not self-intersect
and converge to the segments $s_{i}$ of $\Delta$ with the same endpoints.
(Strictly speaking the convergence property has been stated for bumpy metrics
but it still holds in the flat case since every couple of points of $\Delta$
are connected by a unique geodesic.) Now, we extend this process by deforming
simultaneously every segment $s_{i}$ into a vertex or an edge of the
triangulation through a family of segments whose endpoints move along the
sides of $\Delta$ to the closest vertices of $\Delta$ (see figure 2).

Figure 2.

This deformation $(\delta_{t})$ of $\delta$ can be performed so that the loops
forming the one-cycles $\delta_{t}$ remain simple, except possibly for the
final loop $\delta^{\prime}=\delta_{\infty}$. This defines our deformation
process.

The final one-cycle $\delta^{\prime}$ lies in the one-skeleton of $M_{0}$.
Therefore, the two domains bounded by $\delta_{t}$ converge to two simplicial
domains $D^{\prime}_{1}$ and $D^{\prime}_{2}$ of $M_{0}$ whose common boundary
lies in $\delta^{\prime}$.

######  Lemma 5.2.

We have

${\rm length}(\delta^{\prime})\leq 2\,{\rm length}(\delta).$

###### Proof.

Let $i\in I$. By definition, the arc $\tau_{i}$, of length $\ell_{i}$, lies in
a triangle $\Delta$ of the triangulation of $M_{0}$ and ${\rm
length}(s_{i})\leq\ell_{i}$. Through the deformation process, the segment
$s_{i}$ deforms into either a vertex or an edge of $\Delta$. The latter case
occurs when the two endpoints of $s_{i}$ are at distance less than
$\frac{1}{2}$ from two distinct vertices of $\Delta$. Thus, the segment
$s_{i}$ deforms into a path of length at most twice the length of $s_{i}$.
(The limit case occurs when $s_{i}$ joins the midpoints of two edges of
$\Delta$.) Summing up over all the $i\in I$ yields the desired inequality. ∎

######  Lemma 5.3.

For $j=1,2$, we have

${\rm area}(D^{\prime}_{j})\geq\frac{1}{2}\,{\rm area}(D_{j}).$ (5.5)

###### Proof.

The intersection $D_{j}\cap\Delta$ between $D_{j}$ and a triangle $\Delta$ of
the triangulation of $M_{0}$ is formed of domains bounded by the arcs
$\tau_{i}$ and the edges of $\Delta$. Consider $i\in I$ such that $\tau_{i}$
is an arc of $\Delta$.

Suppose that the length $\ell_{i}$ of $\tau_{i}$ is less than $\frac{1}{2}$.

If the endpoints of $\tau_{i}$ lie in the same edge $e$ of $\Delta$, the arc
$\tau_{i}$ deforms either to this edge or to an endpoint of this edge through
the deformation process previously defined (see figure 2). And so does the
region of $\Delta$ bounded by $\tau_{i}$ and the edge $e$.

If the endpoints of $\tau_{i}$ lie in two adjacent edges $e_{1}$ and $e_{2}$
of $\Delta$, the endpoints of $\tau_{i}$ move to the same vertex $v$ of
$\Delta$ through the deformation process (recall that the length of $\tau_{i}$
is less than $\frac{1}{2}$). Thus, the arc $\tau_{i}$ deforms into this vertex
and the region of $\Delta$ bounded by $\tau_{i}$ and the edges $e_{1}$ and
$e_{2}$ retracts onto the vertex $v$.

In both cases, the area of this region of $\Delta$ bounded by $\tau_{i}$ and
either the edge $e$ or the edges $e_{1}$ and $e_{2}$ is less or equal to
$\frac{3}{2\pi}\ell_{i}^{2}$. This isoperimetric inequality comes from the
standard isoperimetric inequality on the plane combined with a symmetry
argument.

Suppose now that the length $\ell_{i}$ of $\tau_{i}$ is at least
$\frac{1}{2}$. In this case, a rougher estimate holds. Specifically, the area
of each of the two domains of $\Delta$ bounded by $\tau_{i}$ is less than the
area of $\Delta$, i.e., $\frac{\sqrt{3}}{4}$.

Therefore, by substracting up the area change of $D_{j}\cap\Delta$ through the
deformation process for every triangle of the triangulation, we obtain

$\displaystyle{\rm area}(D^{\prime}_{j})$ $\displaystyle\geq$
$\displaystyle{\rm area}(D_{j})-\sum_{i\in
I^{-}}\frac{3}{2\pi}\ell_{i}^{2}-\sum_{i\in I^{+}}\frac{\sqrt{3}}{4}$ (5.6)
$\displaystyle\geq$ $\displaystyle{\rm area}(D_{j})-\sum_{i\in
I^{-}}\frac{3}{4\pi}\ell_{i}-\sum_{i\in I^{+}}\frac{\sqrt{3}}{2}\ell_{i}$
$\displaystyle\geq$ $\displaystyle{\rm
area}(D_{j})-\frac{\sqrt{3}}{2}\left(\sum_{i\in I}\ell_{i}\right)$

where the first sum is over all the indices $i$ for which
$\ell_{i}<\frac{1}{2}$ and the second over those for which
$\ell_{i}\geq\frac{1}{2}$.

From the inequality (5.4), namely

${\rm length}(\delta)=\sum_{i\in I}\ell_{i}<\frac{1}{2}{\rm area}(D_{j}),$

we derive

${\rm area}(D^{\prime}_{j})\geq\frac{1}{2}{\rm area}(D_{j}).$

∎

From the inequality (5.3) and the lemmas 5.2 and 5.3, we deduce

${\rm
length}(\delta^{\prime})<4\,(1+\varepsilon)\,\sqrt{96\pi}\,\sqrt{g+1}\,\frac{\min\\{{\rm
area}(D^{\prime}_{1}),{\rm area}(D^{\prime}_{2})\\}}{\sqrt{{\rm
area}(M_{0})}}.$

When $N\leq N_{*}$, we can take for $D_{1}$ any triangle of $M_{0}$ and for
$D_{2}$ the union of the remaining triangles. Thus,

${\rm length}(\partial D_{i})=3\quad\text{ and }\quad\min\\{{\rm
area}(D_{1}),{\rm area}(D_{2})\\}=\frac{\sqrt{3}}{4}.$

Using the formula (5.2) and the bound $N\leq N_{*}$, one can check that the
inequality (5.1) is satisfies in this case too. Hence the conclusion. ∎

##  6\. Diastole comparison

Let $M_{0}$ be a simplicial closed surface decomposed into two simplicial
domains $D_{1}$ and $D_{2}$ with disjoint interiors. Denote by $\delta$ the
common boundary of the domains $D_{1}$ and $D_{2}$. Consider the abstract
simplicial cones over the connected components of $\partial D_{i}$ composed of
triangles with basis the edges of $\delta=\partial D_{i}$. Denote by $M_{i}$
the simplicial closed surface obtained by attaching these simplicial cones to
$D_{i}$ along $\partial D_{i}$.

The following estimate on the (technical) diastole, cf. Definition 2.2,
follows from a cut-and-paste argument.

######  Proposition 6.1.

We have

${\rm dias}^{\prime}(M_{0})\leq\max_{i}{\rm dias}^{\prime}(M_{i})+{\rm
length}(\delta).$ (6.1)

###### Proof.

Fix $\epsilon>0$. By definition, for $i=1,2$, there exists a one-parameter
family $(z^{i}_{t})_{0\leq t\leq 1}$ of one-cycles on $M_{i}$, with
coefficients in $\Bbbk$, which satisfies the conditions (D.1-5) of the
definition of the (technical) diastole, cf. Definition 2.2, such that

${\rm dias}^{\prime}(M_{i})\leq\sup_{0\leq t\leq 1}\mathbf{M}(z^{i}_{t})<{\rm
dias}^{\prime}(M_{i})+\epsilon.$

Denote by $(\gamma_{j,t}^{i})$ the homotopies of the loops defining the family
$(z_{t}^{i})$, cf. item (D.3) of Definition 2.2. By slightly perturbing the
family $(z^{i}_{t})$ and refining the subdivision $(t_{p})$ if necessary, we
can assume that the loops $\gamma_{j,t}^{i}$ are disjoint or transverse to
$\partial D_{i}$ for every $t\neq t_{p}$, and that they have at most finitely
many tangent points with $\partial D_{i}$ for every $t_{p}$, where
$p=1,\cdots,k$.

The boundary $\partial(z^{i}_{t}\llcorner D_{i})$ of the restriction of
$z^{i}_{t}$ to the domain $D_{i}$ defines a family of zero-cycles on $\partial
D_{i}$ with disjoint supports composed of finitely many points. This family of
one-cycles sweeps out $\partial D_{i}$, that is, it starts and ends at null
one-cycles, and induces a generator of $\pi_{1}(\mathcal{Z}_{0}(\partial
D_{i};\Bbbk),\\{0\\})$. It also induces a family of one-chains
$\alpha^{i}_{t}$ on $\partial D_{i}$ such that $\alpha^{i}_{0}=0$,
$\alpha^{i}_{1}=\pm\partial D_{i}$ and
$\partial\alpha^{i}_{t}=\partial(z^{i}_{t}\llcorner D_{i})$.

In a more constructive way, this family of one-chains on $\partial D_{i}$ can
be described as follows. The restriction $z^{i}_{t}\llcorner D_{i}$ of
$z^{i}_{t}$ to $D_{i}$, with $t_{p}\leq t\leq t_{p+1}$, decomposes as a finite
sum of loops lying in $D_{i}$ plus a finite sum of arcs $c_{j,t}^{i}$ lying in
$D_{i}$ with endpoints on $\partial D_{i}$. As $t$ runs over
$[t_{p},t_{p+1}]$, the basepoints and the endpoints of the arcs $c_{j,t}^{i}$
describe some arcs on $\partial D_{i}$, cf. [Rh55, §14] for a more general
construction. The arcs described by the basepoints (resp. endpoints) of the
$c_{j,t}^{i}$’s inherit a negative (resp. positive) orientation. Note that
these arcs do not overlap because of the condition (D.5). Putting together the
sum of these arcs on $\partial D_{i}$ gives rise to a family of one-chains
$\alpha_{t}^{i}$ on $\partial D_{i}$, starting at the null one-chain and
ending at the one-cycle $\pm\partial D_{i}$, such that
$\partial\alpha^{i}_{t}=\partial(z^{i}_{t}\llcorner D_{i})$.

Consider the family of one-cycles $\tilde{z}^{i}_{t}=(z^{i}_{t}\llcorner
D_{i})-\alpha^{i}_{t}$ lying in $D_{i}$. By construction, this family starts
at the null one-cycle, ends at the one-cycle $\pm\partial D_{i}$ and induces a
generator of $\pi_{1}(\mathcal{Z}_{1}(D_{i},\partial D_{i};\Bbbk),\\{0\\})$.
Furthermore, its mass is bounded as follows

$\mathbf{M}(\tilde{z}^{i}_{t})\leq\mathbf{M}(z^{i}_{t})+{\rm length}(\delta).$

We can now define a new one-parameter family $(\tilde{z}_{t})_{0\leq t\leq 1}$
of one-cycles on $M_{0}$ by concatenating $\tilde{z}^{1}_{t}$ and
$\tilde{z}^{2}_{t}$ as follows

$\tilde{z}_{t}=\left\\{\begin{array}[]{ll}\tilde{z}^{0}_{2t}&\text{if
}t\in[0,1/2],\\\ \\\ -\tilde{z}^{1}_{2-2t}&\text{if }t\in[1/2,1].\\\
\end{array}\right.$

By construction, this new family satisfies the conditions (D.1-4) of the
definition of the (technical) diastole, cf. Definition 2.2, and

$\mathbf{M}(\tilde{z}_{t})<\max_{i}{\rm dias}^{\prime}(M_{i})+{\rm
length}(\delta)+\epsilon.$ (6.2)

It does not necessarily satisfy the condition (D.5) though. Indeed, the
supports of the one-cycles $\tilde{z}_{t}$ may have a nonempty intersection on
$\partial D_{i}$.

By slightly perturbing the family $(\tilde{z}_{t})$ in the neighborhood of
$\partial D_{i}$, it is possible to obtain a family of one-cycles satisfying
the conditions (D.1-5) and the inequality (6.2)

Therefore, ${\rm dias}^{\prime}(M_{0})\leq\max_{i}{\rm
dias}^{\prime}(M_{i})+{\rm length}(\delta)+\epsilon$ for any positive
$\epsilon$. Hence the proposition. ∎

##  7\. Proof of the diastolic inequality

We can now prove the main diastolic inequality, cf. Theorem 1.1. We will use
the notations of the previous sections.

From the discussion in Section 3, cf. Corollary 3.6, it is enough to show that

${\rm dias}^{\prime}(M_{0})\leq\frac{C}{K^{2}}\,\sqrt{g+1}\,\sqrt{{\rm
area}(M_{0})}$ (7.1)

for every simplicial closed surface $M_{0}$ (not necessarily orientable) of
genus $g$. We will argue by induction both on the number $N$ of triangles
composing $M_{0}$ and on its genus. From now on we suppose that Theorem 1.1 is
proved for surfaces of genus less than $g$. If $M_{0}$ is a $2$-sphere, the
proof proceeds by induction only on the number of simplices.

Decompose $M_{0}$ into two simplicial domains $D_{1}$ and $D_{2}$ as in
Proposition 5.1. Consider the simplicial closed surfaces $M_{i}$ defined in
Section 6 by attaching simplicial cones along $\partial D_{i}$. By
construction, the genus of $M_{i}$ is at most $g$. Since $\partial D_{i}$ is
made up of ${\rm length}(\partial D_{i})$ edges, the area of the cones glued
to $D_{i}$ is equal to $\frac{\sqrt{3}}{4}\,{\rm length}(\partial D_{i})$.
Hence the formula

${\rm area}(M_{i})={\rm area}(D_{i})+\frac{\sqrt{3}}{4}{\rm length}(\partial
D_{i}).$ (7.2)

Set $N_{0}:=\sqrt{3}\,C_{0}^{2}\,(g+1)=225\sqrt{3}\cdot 96\pi(g+1).$

######  Lemma 7.1.

For $N\geq N_{0}$, the simplicial closed surfaces $M_{i}$ are composed of
fewer triangles than $M_{0}$.

###### Proof.

Recall that ${\rm area}(M_{0})=\frac{\sqrt{3}}{4}N$. Thus, for $N\geq N_{0}$,
the quantity $\frac{\sqrt{3}}{4}C_{0}\frac{\sqrt{g+1}}{\sqrt{{\rm
area}(M_{0})}}$ is less than $1/2$. From the formula (7.2) and the bound
(5.1), we deduce that

$\displaystyle{\rm area}(M_{i})$ $\displaystyle<$ $\displaystyle{\rm
area}(D_{i})+\min\\{{\rm area}(D_{1}),{\rm area}(D_{2})\\}$ $\displaystyle<$
$\displaystyle{\rm area}(D_{1})+{\rm area}(D_{2}).$

Hence, ${\rm area}(M_{i})<{\rm area}(M_{0})$. ∎

Renumbering the $M_{i}$’s if necessary, we can assume that

${\rm dias}^{\prime}(M_{1})\geq{\rm dias}^{\prime}(M_{2}).$

Let $\alpha$ be a real such that $\alpha\,{\rm area}(M_{0})=\min\\{{\rm
area}(D_{1}),{\rm area}(D_{2})\\}$. Note that $0<\alpha\leq\frac{1}{2}$. The
inequality (5.1) can be written

${\rm length}(\delta)\leq C_{0}\,\alpha\,\sqrt{g+1}\sqrt{{\rm area}(M_{0})},$
(7.3)

where $\delta=\partial D_{i}$.

We have two cases to consider.

First case: Suppose that the number $N$ of triangles of $M_{0}$ is at most
$N_{0}$, where $N_{0}$ is defined in Lemma 7.1. Fix an orientation on the
surface $M_{0}$. This orientation induces an orientation on all the triangles
of $M_{0}$ and on their boundaries. Of course, if the surface is
nonorientable, there is nothing to do. Consider the boundaries of the
triangles of $M_{0}$ as one-cycles with coefficients in $\Bbbk$. The sum of
these boundaries, viewed as one-cycles, vanishes. Given a triangle $\Delta$ of
$M_{0}$, there exists a homotopy from the boundary of $\Delta$, with the
induced orientation, to the center of $\Delta$ composed of disjoint loops. The
sum of these homotopies over all the triangles of $M_{0}$ defines a one-
parameter family of one-cycles $(z_{t})_{t}$ on $M_{0}$, which starts and ends
at the null one-cycle. Furthermore, it induces a generator of
$\pi_{1}(\mathcal{Z}_{1}(M_{0};\Bbbk),\\{0\\})\simeq H_{2}(M_{0};\Bbbk)$ and
satisfies the technical conditions (D.3-5) of Definition 2.2. Therefore,

${\rm dias}^{\prime}(M_{0})\leq\sup_{t}\mathbf{M}(z_{t})\leq 3N.$

Since ${\rm area}(M_{0})=\frac{\sqrt{3}}{4}N$ and $N\leq N_{0}$, we obtain

$\displaystyle{\rm dias}^{\prime}(M_{0})$ $\displaystyle\leq$ $\displaystyle
3\sqrt{N}\sqrt{N_{0}}$ $\displaystyle\leq$ $\displaystyle 2\cdot
3^{3/4}\,\sqrt{N_{0}}\sqrt{{\rm area}(M_{0})}$ $\displaystyle\leq$
$\displaystyle 6\,C_{0}\,\sqrt{g+1}\,\sqrt{{\rm area}(M_{0})}.$

Second case: Suppose that $N$ is greater than $N_{0}$. We can assume that

$\frac{{\rm dias}^{\prime}(M_{1})}{\sqrt{{\rm area}(M_{1})}}\leq\frac{{\rm
dias}^{\prime}(M_{0})}{\sqrt{{\rm area}(M_{0})}}$

otherwise the diastolic inequality (7.1) follows from Lemma 7.1 by induction
on $g$ and $N$. That is,

${\rm dias}^{\prime}(M_{1})\leq\lambda\,{\rm dias}^{\prime}(M_{0}),$

where

$\lambda:=\sqrt{\frac{{\rm area}(M_{1})}{{\rm area}(M_{0})}}.$ (7.4)

Note that $\lambda<1$ from Lemma 7.1. From Proposition 6.1 and since ${\rm
dias}^{\prime}(M_{1})=\max_{i}{\rm dias}^{\prime}(M_{i})$, we deduce

$\displaystyle{\rm dias}^{\prime}(M_{0})$ $\displaystyle\leq$
$\displaystyle\frac{1}{1-\lambda}\,{\rm length}(\delta).$

From the inequality (7.3), we obtain

${\rm dias}^{\prime}(M_{0})\leq
C_{0}\,\frac{\alpha}{1-\lambda}\,\sqrt{g+1}\,\sqrt{{\rm area}(M_{0})}.$ (7.5)

######  Lemma 7.2.

We have $\displaystyle\frac{\alpha}{1-\lambda}\leq 4$.

###### Proof.

Since $N\geq N_{0}$, the quantity
$\frac{\sqrt{3}}{4}C_{0}\frac{\sqrt{g+1}}{\sqrt{{\rm area}(M_{0})}}$ is less
than $\frac{1}{2}$ and

${\rm area}(M_{i})\leq{\rm area}(D_{i})+\frac{1}{2}\min\\{{\rm
area}(D_{1}),{\rm area}(D_{2})\\}$

from the formula (7.2) and the bound (5.1).

If ${\rm area}(D_{1})\leq{\rm area}(D_{2})$, then ${\rm
area}(D_{1})=\alpha\,{\rm area}(M_{0})$. In this case, we deduce that
$\lambda\leq\sqrt{\frac{3}{2}\alpha}$ from the definition (7.4) of $\lambda$.
Since $\alpha\leq\frac{1}{2}$, we obtain

$\frac{\alpha}{1-\lambda}\leq\frac{\alpha}{1-\sqrt{\frac{3}{2}\alpha}}\leq
2+\sqrt{3}.$

Otherwise, ${\rm area}(D_{1})=(1-\alpha)\,{\rm area}(M_{0})$. As previously,
we derive, in this case, that $\lambda\leq\sqrt{1-\frac{\alpha}{2}}$. Since
$\alpha>0$, we get

$\frac{\alpha}{1-\lambda}\leq 2\left(1+\sqrt{1-\frac{\alpha}{2}}\right)\leq
4.$

∎

This lemma with the inequality (7.5) implies that

${\rm dias}^{\prime}(M_{0})\leq 4\,C_{0}\,\sqrt{g+1}\,\sqrt{{\rm
area}(M_{0})}.$

Therefore, the desired diastolic inequality (7.1) holds in both cases.

######  Remark 7.3.

Let $(z_{t})$ be a family of one-cycles which sweeps out a closed Riemannian
surface $M$, cf. (D.1-2). From the constructions of [Al60], there exists
$t_{0}$ such that $z_{t_{0}}$ decomposes $M$ into two disjoint domains of area
at least $\frac{1}{4}{\rm area}(M)$. Hence,

${\rm dias}(M)\geq\mathbf{M}(z_{t_{0}})\geq\frac{1}{4}h(M){\rm area}(M).$

The closed hyperbolic surfaces of arbitrarily large genus with Cheeger
constant bounded away from zero constructed in [Br86] provide examples of
surfaces with ${\rm area}(M)\simeq g$ and ${\rm dias}(M)\gtrsim g$. Thus, the
dependence of the inequality (1.5) on the genus is optimal.

##  8\. Comparison of Riemannian invariants on the two-sphere

In this section, we clarify the relationships between several Riemannian
invariants on the two-sphere by comparing them up to an equivalence relation.

######  Definition 8.1.

Two Riemannian invariants $I_{1}$ and $I_{2}$ are said to be _essentially
equivalent_ on a manifold $M$ if there exist two positive constants $C$ and
$C^{\prime}$ such that for every Riemannian metric ${\mathcal{G}}$ on $M$, we
have

$C\,I_{1}({\mathcal{G}})\leq I_{2}({\mathcal{G}})\leq
C^{\prime}\,I_{1}({\mathcal{G}}).$

We denote by $I_{1}(M)\simeq I_{2}(M)$ this equivalence relation.

Together with the filling radius introduced in Definition 4.3, we will
consider the following Riemannian invariants.

######  Definition 8.2.

Let $M$ be a closed $n$-dimensional Riemannian manifold.

  * •

The _$1$ -hypersphericity_, denoted by ${\rm HS}(M)$, is the supremum of the
positive reals $R$ so that there is a contracting map of degree $1$ from $M$
to the $n$-sphere of radius $R$.

  * •

The _Uryson $1$-width_, denoted by ${\rm UW}_{1}(M)$, is the infimum of the
positive reals $W$ so that there is a continuous map from $M$ to a
$1$-dimensional polyhedron whose fibers have diameter less than $W$.

  * •

Let us define the functional $\mu$ on the one-cycle space
$\mathcal{Z}_{1}(M,{\mathbb{Z}})$ of a Riemannian two-sphere $M$ as

$\mu(z)=\sup\\{\mathbf{M}(z_{i})\mid z_{i}\text{ is a connected component of
}z\\}.$

We define a critical value of the functional $\mu$ by a global minimax
principle

$L(M)=\inf_{(z_{t})}\sup_{0\leq t\leq 1}\mu(z_{t})$

where $(z_{t})$ runs over the families of one-cycles satisfying the following
conditions:

    1. (C.1)

$z_{t}$ starts and ends at null one-cycles;

    2. (C.2)

$z_{t}$ induces a generator of $H_{2}(M;{\mathbb{Z}})$.

In the following proposition, we compare these Riemannian invariants on the
two-sphere.

######  Proposition 8.3.

The hypersphericity, the Uryson $1$-width, the filling radius and $L$ are
essentially equivalent on the two-sphere $S^{2}$, that is

${\rm UW}_{1}(S^{2})\simeq{\rm HS}(S^{2})\simeq{\rm FillRad}(S^{2})\simeq
L(S^{2}).$

###### Proof.

Since the filling radius of every closed Riemannian $n$-manifold decreases
under contracting maps of degree one, cf. [Gr83], we have

${\rm FillRad}(S^{2},{\rm can})\,{\rm HS}(M)\leq{\rm FillRad}(M)$ (8.1)

where the value of the filling radius of the standard spheres has been
computed in [Ka83]. Here, ${\rm FillRad}(S^{2},{\rm
can})=\frac{1}{2}\arccos\left(-\frac{1}{3}\right)$.

The inequality

${\rm UW}_{1}(M)\leq{\rm HS}(M)$ (8.2)

holds for every closed Riemannian manifold, cf. [Gr88].

The inequalities

$\frac{L(M)}{120}\leq{\rm HS}(M)\leq\frac{2}{\pi}{\rm UW}_{1}(M)$ (8.3)

have been established in [Gu05, Theorem 0.3 and p. 1058].

Next we will prove that there exists a constant $C$

${\rm FillRad}(M)\leq C\,L(M).$ (8.4)

First of all observe that if $(z_{t})$ is a family of one-cycles on $M$ and if
$t_{1}$ and $t_{2}$ are close enough, the one-cycle $z=z_{t_{2}}-z_{t_{1}}$
bounds a $2$-chain of small mass on $M$. Indeed, by definition of the flat
norm, cf. Section 2, and since the one-cycle $z$ has a small flat norm, there
exists a $2$-chain $A$ such that both $\mathbf{M}(z-\partial A)$ and
$\mathbf{M}(A)$ are small. By the isoperimetric inequality on $M$, there
exists a $2$-chain $B$ of small mass bounding the one-cycle $z-\partial A$.
Thus, the $2$-chain $A+B$ bounds $z$ and has a small mass. Now, using the
arguments of [Gr83, p. 133], one can show that, for every $\varepsilon>0$,
there exists a $1$-Lipschitz map $\partial X\to M$ of degree one, where $X$ is
a $3$-dimensional pseudomanifold endowed with a metric length structure such
that $d(x,\partial X)\leq C\,L(M)+\varepsilon$. We immediately derive the
inequality (8.4).

Putting together the inequalities (8.1), (8.2), (8.3) and (8.4) yields the
desired relations. ∎

######  Remark 8.4.

We have $L(M)\leq{\rm dias}(M)$ for every Riemannian two-sphere $M$ but the
diastole is _not_ essentially equivalent to any of the Riemannian invariant of
Proposition 8.3 on the two-sphere from [Sa04, Theorem 1.6]. In the appendix,
we bound from below the filling radius of the Riemannian two-spheres in terms
of the invariant $L$ using different methods.

## Appendix

In this appendix, we prove the following result which first appeared in [Sa01,
§2.4.2] and was not published in a journal.

######  Proposition A.1.

Let $M$ be a Riemannian two-sphere, then

$L(M)\leq 18\,{\rm FillRad}(M).$

######  Remark A.2.

The multiplicative constant in this inequality is better than the one obtained
in the proof of Proposition 8.3. Furthermore, it follows from a more general
result, namely Proposition A.4.

From the bounds ${\rm FillRad}(M)\leq\sqrt{{\rm area}(M)}$, cf. [Gr83], and
${\rm FillRad}(M)\leq\frac{1}{3}{\rm diam}(M)$, cf. [Ka83], we immediately
deduce the following corollary.

######  Corollary A.3.

Let $M$ be a Riemannian two-sphere, then

$\displaystyle L(M)$ $\displaystyle\leq 18\,\sqrt{{\rm area}(M)}$ (A.1)
$\displaystyle L(M)$ $\displaystyle\leq 6\,{\rm diam}(M).$ (A.2)

In order to prove Proposition A.1, we need to extend the definition of the
invariant $L$, cf. Definition 8.2, as follows.

Let $N\subset M$ be a connected domain whose boundary $\partial N$ is a finite
union of closed geodesics $c_{i}$ with $i\in I$.

Let us consider the one-parameter families of one-cycles $(z_{t})_{0\leq t\leq
1}$ on $N$ which satisfy the following conditions.

  1. (C’.1)

$z_{0}=\cup_{i\in I_{0}}c_{i}$ and $z_{1}=\cup_{i\in I_{1}}c_{i}$ for some
partition of $I=I_{0}\amalg I_{1}$

  2. (C’.2)

$z_{t}$ induces a generator of $H_{2}(N,\partial N;{\mathbb{Z}})$.

If $I_{0}$ or $I_{1}$ is empty, the corresponding one-cycles are reduced to
the null one-cycle.

We define a critical value of the functional $\mu$ by a global minimax
principle

$L(N)=\inf_{(z_{t})}\sup_{0\leq t\leq 1}\mu(z_{t})$

where $(z_{t})$ runs over the families of one-cycles satisfying the conditions
(C’.1-2) above.

We will need a few more definitions. Every nontrivial simple loop $\gamma$ of
$M$ admits two sides defined as the connected components of
$M\setminus\gamma$. A nontrivial simple geodesic loop $\gamma$ is said to be
minimizing with respect to one of its sides if every pair of points of
$\gamma$ can be joined by a minimizing segment which does not pass through
this side. A simple geodesic loop minimizing with respect to both sides is
said to be minimizing.

Now, we can state the following result which immediately leads to Proposition
A.1 when $N=M$ and $\partial N=\emptyset$.

######  Proposition A.4.

Let $M$ be a Riemannian two-sphere and $N\subset M$ be a connected domain
whose boundary $\partial N$ is a finite union of minimizing geodesic loops
$c_{i}$. Then

${\rm FillRad}(M)\geq\frac{1}{18}L(N).$

Without loss of generality, we can assume the metric on $M$ is bumpy (a
generic condition). The curve-deformation process described below plays a key
role in the proof of the above proposition.

Let $\gamma$ be a non-minimizing simple geodesic loop on $M$ and $c$ be a
minimizing arc intersecting $\gamma$ only at its endpoints. The arc $c$
divides $\gamma$ into $c^{+}$ and $c^{-}$. The simple loops $c^{+}\cup c$ and
$c^{-}\cup c$, endowed with the orientations induced by $\gamma$, bound convex
domains and converge to simple geodesic loops $\gamma^{+}$ and $\gamma^{-}$
through homotopies $z^{+}_{t}$ and $z^{-}_{t}$ given by the curvature flow.
Applying the curve-shortening process of [Sa04, §2] if necessary, we can
extend the homotopies and suppose that $\gamma^{+}$ and $\gamma^{-}$ are local
minima of the mass functional. Since the curves $c^{+}\cup c$ and $c^{-}\cup
c$ do not intersect transversely and bound convex domains, the simple loops
$z^{+}_{t}$ and $z^{-}_{t}$ are disjoint for $t>0$. The sum
$z_{t}=z^{+}_{t}+z^{-}_{t}$ defines the curve-deformation process of $\gamma$
with respect to $c$. The homotopy of the one-cycles $z_{t}$ is
$\mu$-nonincreasing and satisfies $\mu(z_{t})\leq{\rm length}(\gamma)$. In
particular, ${\rm length}(\gamma^{+})$ and ${\rm length}(\gamma^{-})$ are less
than ${\rm length}(\gamma)$.

In the following two lemmas, we assume that the only minimizing geodesic loops
of $N$ are the connected components of the boundary $\partial N$.

######  Lemma A.5.

Let $\gamma$ and $\gamma^{\prime}$ be two disjoint (nontrivial) simple
geodesic loops of $N$. Suppose that $\gamma$ (resp. $\gamma^{\prime}$) is
minimizing with respect to the side which does not contain $\gamma^{\prime}$
(resp. $\gamma$).
Then, $\gamma$ or $\gamma^{\prime}$ is a connected component of $\partial N$.

###### Proof.

The geodesics $\gamma$ and $\gamma^{\prime}$ bound a connected domain
$N^{\prime}\subset N$ with $\gamma,\gamma^{\prime}\subset\partial N^{\prime}$.
Suppose that $\gamma$ does not lie in $\partial N$. Since the only minimizing
geodesic loops of $N$ are the connected components of $\partial N$, the loop
$\gamma$ is not minimizing. Therefore, there exists a minimizing arc $c$ in
$N^{\prime}$ with endpoints in $\gamma$ which does not lie in $\gamma$. The
curve-deformation process applied to $\gamma$ with respect to $c$ yields two
disjoint simple geodesic loops of $N^{\prime}$. The union of these two
geodesic loops is noted $z_{1}$. Let us define by induction on $k$ a sequence
$z_{k}$ formed of a disjoint union of simple geodesic loops of $N^{\prime}$.
This sequence of one-cycles is $\mu$-nonincreasing. Furthermore, each $z_{k}$
bounds with $\gamma$ a connected domain $D_{k}$ of $N^{\prime}$ with $\partial
D_{k}=z_{k}\cup\gamma$. If one of the geodesic loops of $z_{k}$ is not
minimizing with respect to the side opposed to $\gamma$, we apply to it the
curve-deformation process with respect to this side. Otherwise, if one of them
is not minimizing with respect to the side containing $\gamma$, we apply to it
the curve-deformation process. In both cases, we obtain a new collection
$\mathcal{C}$ of simple geodesic loops which do not intersect each other. We
define $z_{k+1}$ as the union of the nontrivial geodesic loops of
$\mathcal{C}$ which can be joined to $\gamma$ by paths cutting no loop of
$\mathcal{C}$. This induction process stops after a finite number $n$ of steps
when $z_{n}$ is only formed of minimizing geodesic loops of $N^{\prime}$.
Since there is no minimizing geodesic loop in the interior of $N$, we have
$z_{n}\subset\partial N$. Therefore, the geodesic loop $\gamma^{\prime}$,
which lies between $z_{n}$ and $\partial N$, is a connected component of
$\partial N$. ∎

######  Lemma A.6.

Let $\gamma$ be a simple geodesic loop of $N$ of length less than $L(N)$ and
suppose that $\gamma$ is minimizing with respect to no side.
There exists a non-minimizing simple geodesic loop of $N$, lying in a side of
$\gamma$, which is minimizing with respect to the side opposed to $\gamma$.

###### Proof.

Assume that every simple geodesic loop of $N$, disjoint from $\gamma$ and
minimizing with respect to the side opposed to $\gamma$, is minimizing. Let us
recall that the only minimizing geodesic loop of $N$ are the connected
components of $\partial N$. We apply the curve-deformation process to $\gamma$
with respect to each of its sides $N_{1}$ and $N_{2}$. Then, we extend these
homotopies as in the proof of Lemma A.5. By assumption, one of these
homotopies lies in $N_{1}$ and the other in $N_{2}$. Since there is no
minimizing geodesic loop in the interior of $N$, these homotopies, put
together, yield a one-parameter family of one-cycles $z_{t}$ in $N$, which
starts from $\partial N_{1}\cap\partial N$, passes through $\gamma$ and ends
at $\partial N_{2}\cap\partial N$. Furthermore, the family $(z_{t})$ induces a
generator of $H_{2}(N,\partial N)$ and satisfies $\mu(z_{t})\leq{\rm
length}(\gamma)$. Therefore, ${\rm length}(\gamma)\geq L(N)$ from the
definition of $L(N)$. ∎

######  Remark A.7.

From Lemma A.5, the non-minimizing simple geodesic loops of Lemma A.6 lie in
only one side of $\gamma$.

Using the lemma A.6 and the remark A.7, we can define the side of deformation
of a geodesic loop and make the curve-deformation process more precise.

######  Definition A.8.

The side of deformation of a non-minimizing simple geodesic loop $\gamma$ of
$N$ of length less than $L(N)$ is

  1. (1)

the opposite side with respect to which $\gamma$ is minimizing, if $\gamma$ is
minimizing with respect to a side.

  2. (2)

the side which does not contain any geodesic loop minimizing with respect to
the side opposed to $\gamma$ (except the connected components of $\partial
N$), otherwise.

From now on, when we will apply the curve-deformation process to such a
geodesic loop $\gamma$, it will always be with respect to a minimizing arc
with endpoints in $\gamma$ lying in the deformation side of $\gamma$.

###### Proof of Proposition A.4.

Let us assume first that the only minimizing geodesic loops of $N$ are the
connected components of $\partial N$. We argue by contradiction as in the
proof of Proposition 4.4. Pick two reals $r$ and $\varepsilon$ such that ${\rm
FillRad}(M)<r<r+\varepsilon<\frac{1}{8}L(M)$. Let $i:M\hookrightarrow
L^{\infty}(M)$ be the Kuratowski distance preserving embedding defined by
$i(x)(.)=d_{M}(x,.)$. By definition of the filling radius, there exists a map
$\sigma:P\to U_{r}$ from a $3$-complex $P$ to the $r$-tubular neighborhood
$U_{r}$ of $i(M)$ in $L^{\infty}=L^{\infty}(M)$ such that $\sigma_{|\partial
P}:\partial P\to i(M)$ represents the fundamental class of $i(M)\simeq M$ in
$H_{2}(i(M);{\mathbb{Z}})\simeq H_{2}(M;{\mathbb{Z}})$. Let us show that the
map $\sigma_{|\partial P}:\partial P\to i(M)$ extends to a map $f:P\to i(M)$.

Deforming $\sigma$ and subdividing $P$ if necessary, we define an extension
$f:P^{1}\cup\partial P\to i(M)\simeq M$ of $\sigma_{|\partial P}$ to the
$1$-skeleton of $P$ as in the proof of Proposition 4.4. From now on, we will
identify $i(M)$, $i(N)$ and $i(\partial N)$ with $M$, $N$ and $\partial N$.
Recall that the image under $f$ of the boundary of every $2$-simplex of $P$ is
a simple loop of length less than $3\rho$, where $\rho=2r+\varepsilon$. Since
the connected components of $\partial N$ are minimizing geodesic loops, the
images of the edges of $P$ cut $\partial N$ at most twice. In particular, the
images of the boundary of the $2$-simplices of $P$ cut $\partial N$ at most
six times.

On every edge of $P$ whose image cuts $\partial N$, we introduce new vertices
given by the preimages of the intersection points with $\partial N$. Let
$\Delta^{2}$ be a $2$-simplex of $P$. Every pair of new vertices of
$\Delta^{2}$ which map to the same connected component of $\partial N$ defines
a new edge in $\Delta^{2}$. By definition, the images by $f$ of these new
edges are the minimizing segments (lying in $\partial N$) joining the images
of their endpoints. We number the vertices of $P$, old and new, and consider
the natural induced order. The new edges of $\Delta^{2}$, which map to $N$,
bound a domain $D^{2}$ of $\Delta^{2}$. We introduce new edges on $D^{2}$
joining the greatest vertex of $D^{2}$ to every other vertex of $D^{2}$. This
decomposes $\Delta^{2}$ into triangles. By definition, each new edge maps onto
a minimizing segment of $N$ joining the images of its endpoints. The triangles
of the faces of every $3$-simplex $\Delta^{3}$ whose boundaries map to $N$
bound a domain $D^{3}$ of $\Delta^{3}$. We introduce, as previously, new edges
on $D^{3}$ joining the greatest vertex of $D^{3}$ to every other vertex of
$D^{3}$. These new edges define new faces in $\Delta^{3}$. These new faces
decompose $\Delta^{3}$ into $3$-simplices. As previously, each new edge maps
onto a minimizing segment of $N$ joining the images of its endpoints.

In conclusion, we have defined a new simplicial structure on $P$, a map
$f:P^{1}\cup\partial P\longrightarrow M$ with $f_{|\partial
P}=\sigma_{|\partial P}$ and a decomposition of $P$ into two finite
subcomplexes $P^{\prime}$ and $P^{\prime\prime}$ such that

  1. (1)

the image of the 1-skeleton of every 3-simplex of $P^{\prime}$ (resp.
$P^{\prime\prime}$) lies in the closure of $N$ (resp. $M\setminus N$).

  2. (2)

the boundary of every 2-simplex of $P^{\prime}$ maps into a geodesic triangle
whose length of the edges is less than $\frac{3}{2}\rho<\frac{1}{6}L(N)$

We want to extend $f$ to $P^{2}$. Let $\partial\Delta^{2}$ be the boundary of
a $2$-simplex $\Delta^{2}$ of $P$ which does not lie in $\partial P$. If
$\partial\Delta^{2}$ lies in $P^{\prime\prime}$, we fill its image by the disk
it bounds in $M\setminus N$. Otherwise, the image of $\partial\Delta^{2}$ lies
in $N$. In this case, it converges to a simple geodesic loop $c$ of $N$ by the
curve-shortening process of [Sa04, §2]. We want now to define a
$\mu$-nonincreasing homotopy of one-cycles in $N$ from $c$ to some connected
components of $\partial N$, which gives rise to a filling of the image of
$\partial\Delta^{2}$ in $M$.

The images of the boundaries of the $2$-simplices of $P^{\prime}$ converge to
simple geodesic loops of $N$. The set formed of these simple geodesic loops
which are not minimizing is noted $\mathcal{C}$. It is finite since
$P^{\prime}$ is a finite simplicial complex. We want to define homotopies of
one-cycles in $N$ from every loop of $\mathcal{C}$ to some minimizing geodesic
loops. Let us consider the deformation sides of two disjoint geodesic loops of
$\mathcal{C}$. From Lemma A.5, either one is contained in the other or their
intersection is disjoint. Therefore, there exists $\gamma_{0}\in\mathcal{C}$
such that no geodesic loop of $\mathcal{C}$ lies in the deformation side of
$\gamma_{0}$. The curve-deformation process applied to $\gamma_{0}$ yields a
homotopy of one-cycles $z_{t}$ defined for $0\leq t\leq 1$ between
$\gamma_{0}$ and the disjoint union $\gamma^{+}_{0}\cup\gamma^{-}_{0}$ of two
simple geodesic loops. The flow of every other loop of $\mathcal{C}$ is
constant for $0\leq t\leq 1$. We repeat this construction with the set of the
non-minimizing geodesic loops of
$\\{\gamma\mid\gamma\in\mathcal{C},\gamma\neq\gamma_{0}\\}\cup\\{\gamma^{+}_{0},\gamma^{-}_{0}\\}$.
Since for a (generic) bumpy metric there are only finitely many geodesic loops
of length uniformly bounded, this process stops after a finite number of
iterations. From Lemma A.5, the deformation side of every simple geodesic loop
of $N$ lying in the deformation side of a geodesic loop $\gamma$ does not
contain $\gamma$. Therefore, the iterations of this process give rise to
$\mu$-nonincreasing homotopies of one-cycles between the geodesic loops
$\gamma$ of $\mathcal{C}$ and the connected components of $\partial N$ which
lie in the deformation side of $\gamma$. A contraction of these latter into
points in $M\setminus N$ yields a filling of the boundary of the $2$-simplices
of $P^{\prime}$ in $M$. Furthermore, the homotopies arising from two geodesic
loops of $\mathcal{C}$ which do not intersect each other remain disjoint at
every time $t$, except possibly for some connected components lying in
$\partial N$. Thus, the sum $z_{t}$ of the homotopies arising from two
disjoint geodesics $\gamma$ and $\gamma^{\prime}$ of $\mathcal{C}$ satisfies
$\mu(z_{t})\leq{\rm length}(\gamma)+{\rm length}(\gamma^{\prime})\leq 9\rho$.

In conclusion, for every $2$-simplex $\Delta^{2}$ of $P$, we get a degree one
map from $\Delta^{2}$ onto its image in $i(M)\simeq M$ which agrees with $f$
on the boundary. This yields the desired extension $f:P^{2}\longrightarrow
i(M)\simeq M$.

Let us extend this map to $P^{3}$. Let $\Delta^{3}$ be a $3$-simplex of $P$.
The restriction of $f$ to the boundary $\partial\Delta^{3}$ is noted
$\varphi:\partial\Delta^{3}\longrightarrow M$. If $\Delta^{3}$ lies in
$P^{\prime\prime}$, it extends to a map defined on $\Delta^{3}$ whose image
lies in the closure of $M\setminus N$. Otherwise, $\Delta^{3}$ lies in
$P^{\prime}$ and the map $\varphi$ induces a class $[\partial\Delta^{3}]$ in
$H_{2}(M,M\setminus N)\simeq H_{2}(N,\partial N)$. This class arises from
degree one maps defined on the faces of $\Delta^{3}$. Therefore, it is trivial
or generates $H_{2}(N,\partial N)$. The map $\varphi$ extends to
$\Delta^{3}\longrightarrow M$ if and only if the class $[\partial\Delta^{3}]$
vanishes in $H_{2}(N,\partial N)$. Since the images of the edges of
$\Delta^{3}$ are minimizing segments, the image of the $1$-skeleton of
$\Delta^{3}$ is isotopic to one of the following two graphs on $M$ shown in
Figure 3.

Therefore, the faces of $\Delta^{3}$ decompose into two pairs such that the
edges of the faces of each pair do not intersect each other transversely. By
construction, the images of the faces of $\partial\Delta^{3}$ are defined by
$\mu$-nonincreasing homotopies of one-cycles $z_{i}^{t}$ between the images of
their boundaries and some connected components of $\partial N$ or null one-
cycles. Furthermore, the sum of the two homotopies $z_{1}^{t}$ and $z_{2}^{t}$
arising from a pair of faces that do not intersect each other transversely
satisfies $\mu(z_{1}^{t}+z_{2}^{t})\leq 9\rho$. The same goes for the other
pair of faces, which yields the homotopies $z_{3}^{t}$ and $z_{4}^{t}$.
Putting together the homotopies $z_{1}^{t}+z_{2}^{t}$ and
$z_{3}^{t}+z_{4}^{t}$, we obtain a one-parameter family of one-cycles $z_{t}$.
By construction, the family $(z_{t})$ represents $[\partial\Delta^{3}]$ and
satisfies $\mu(z_{t})\leq 9\rho<L(N)$. Therefore, the class
$[\partial\Delta^{3}]$ vanishes in $H_{2}(N,\partial N)$.

Thus, the map $f$ extends to each simplex of $P$ and gives rise to an
extension $f:P\longrightarrow i(M)\simeq M$ of $\sigma_{|\partial P}$. This
proves the inequality when $N$ has no minimizing geodesic in its interior.

Suppose now that $N$ admits a minimizing geodesic loop $\gamma$ which does not
lie in $\partial N$. This simple loop decomposes $N$ into two connected
components $N_{1}$ and $N_{2}$, whose boundaries are finite unions of
minimizing simple geodesic loops. By induction on the number of nontrivial
minimizing geodesic loops which lie in the interior of $N$, we have ${\rm
FillRad}(M)\geq\frac{1}{18}L(N_{i})$ for $i=1,2$. Let us consider some
homotopies of one-cycles on $N_{i}$ between the connected components of
$\partial N_{i}$ with $i=1,2$ which induce generators of $H_{2}(N_{i},\partial
N_{i})$ for $i=1,2$. Put together, these homotopies yield one-parameter
families of one-cycles satisfying (C.1-2). We deduce that
$L(N)\leq\max\\{L(N_{1}),L(N_{2})\\}$. This proves the result by induction. ∎

Figure 3.

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