# Bounds on saddle connections for flat spheres

Kai Fu  Institut de Mathématiques de Bordeaux, Talence, France
<EMAIL_ADDRESS>and  Guillaume Tahar  Yanqi Lake Beijing Institute
of Mathematical Sciences and Applications, Huairou District, Beijing, China
<EMAIL_ADDRESS>
###### Abstract.

We consider a flat metric with conical singularities on the sphere. Assuming
no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of
immersed disks to obtain an explicit upper bound on the number of saddle
connections with at most $k$ self-intersections. We also deduce a bound on
their lengths for a surface with a normalized area.

###### Key words and phrases:

Flat metric, Conical singularities, Saddle connection, Delaunay triangulation,
Curvature gap, Flat annulus

###### Contents

  1. 1 Introduction
  2. 2 Discrete curvature
  3. 3 Flat annuli
  4. 4 Geometry of immersed disks
  5. 5 Upper bounds

##  1\. Introduction

A flat sphere is a topological sphere endowed with flat metric in the
complement of finitely many conical singularities. Equivalently, it can be
described by a collection of Euclidean triangles glued isometrically along a
pairing of edges (see [8] for details). When conical angles are rational
multiples of $\pi$, flat metrics on the sphere are induced by
$k$-differentials (see [2] for an extensive study of these differentials and
their moduli spaces).

######  Definition 1.1.

A trajectory in a flat sphere is a geodesic without singularities in the
interior. It is simple if there is no self-intersections in the interior. A
saddle connection is a finite trajectory whose endpoints are conical
singularities. Self-intersections are counted with their multiplicities (see
Definition 5.5).

In this paper, we are interested in the counting of saddle connections on flat
spheres. The key ingredients of our bounds are:

  1. (1)

the notion of the curvature gap, measuring the obstruction to realize a
partition of the set of conical singularities in two sets of equal total
curvatures, see Section 2;

  2. (2)

the geometric properties of flat annuli, analogs of flat cylinders of
translation surfaces, see Section 3;

  3. (3)

locally isometric immersions of disks and subsequent Delaunay triangulations,
see Section 4.

We focus on the sphere case only in this paper because there is no a
reasonable generalisation for our notion of the curvature gap in higher genus
cases.

######  Theorem 1.2.

In a flat sphere with $n$ conical singularities and a curvature gap of
$\delta>0$, the number of saddle connections with at most $k$ self-
intersections is at most $(3n-6)2^{s}$ where
$s=\frac{20n(n-1)\sqrt{k}+20n}{\delta}$.

In the case of simple saddle connections, the upper bound can be improved to
$\frac{1}{(3n-7)!}(\frac{5n}{\delta}+3n-7)^{3n-6}+3n-6$.

Theorem 1.2 is proved in Section 5.2. It follows essentially from an estimate
on combinatorial lengths of simple trajectories in a flat sphere (see
Proposition 5.2) and some topological considerations about self-intersecting
arcs.

According to the counting of simple saddle connections in flat annuli (see
Section 3.3), we emphasise that the curvature gap $\delta$ cannot be
eliminated from the upper bound in Theorem 1.2.

Using an estimate on lengths of edges for Delaunay triangulations of flat
spheres (see Lemma 4.3), we obtain an upper bound on lengths of these
trajectories (including saddle connections). In particular, it does not depend
on the systole111The systole is the length of the shortest simple saddle
connection. of the surface. Delaunay triangulations have already been used to
study distributions of saddle connections in flat surfaces, see [1] for
example.

######  Theorem 1.3.

In a flat sphere of a unit area with $n$ conical singularities and a curvature
gap $\delta>0$, the metric length of any trajectory with at most $k$ self-
intersections is at most
$\frac{40n(n-1)\sqrt{k}+40n}{\delta\sqrt{\pi}}+\frac{20n(n-1)\sqrt{k}+20n}{\delta^{3/2}\sqrt{2\pi}}$.

The bound in the case of a simple trajectory can be slightly improved to
$\frac{10n}{\delta\sqrt{\pi}}+\frac{5n}{\delta^{3/2}\sqrt{2\pi}}$.

Theorem 1.3 is proved in Section 5.3.

We do not discuss the sharpness of our bounds in Theorem 1.2 and Theorem 1.3
in this paper. How to construct sharp bounds will be an interesting question.

The estimate on lengths of Delaunay edges also implies an essentially sharp
upper bound on the diameter of flat spheres (see Section 4.2).

######  Corollary 1.4.

In a flat sphere $X$ of a unit area with $n$ conical singularities and a
curvature gap $\delta>0$, the diameter of $X$ as a metric space is at most
$(n+1)(\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}})$.

#### Acknowledgements.

The first author would like to thank his advisors, Vincent Delecroix and Elise
Goujard, for introducing him to the flat geometry and many valuable
discussions. He wants to acknowledge Yitwah Cheung for his invitation and the
opportunity to conduct this research at Tsinghua University. The authors would
also like to acknowledge Dmitri Panov for valuable remarks.

##  2\. Discrete curvature

For a conical singularity $M$ of an angle $\theta_{M}$ in a flat sphere $X$,
we define its discrete curvature $k_{M}=\frac{2\pi-\theta}{2\pi}$. In
particular we have $k_{M}\in]-\infty,1[$. The classical Gauss-Bonnet formula
writes $\sum\limits_{M\in X}k_{M}=2$.

###  2.1. Curvature gap

The obstruction to realize a partition of the set of conical singularities in
two sets of equal total curvature is measured by the curvature gap.

######  Definition 2.1.

For a flat sphere $X$ with a finite set $\Lambda$ of conical singularities, we
define the curvature gap of $X$ as
$\inf\limits_{I\subset\Lambda}|1-\sum\limits_{i\in I}k_{i}|$.

A combinatorial analog of the curvature gap has been introduced in Section 6
of [6] to characterize strata of quadratic differentials where the
corresponding half-translation surfaces can have infinitely many saddle
connections.

Indeed, any simple closed geodesic cuts out the sphere into two connected
components, each containing a total discrete curvature of $1$ by Gauss-Bonnet
formula. It follows that a flat sphere satisfying $\delta>0$ does not contain
any simple closed geodesic (and therefore no flat cylinder).

###  2.2. Upper bound on the curvature gap

In this Section, we prove a sharp bound on the curvature gap and characterize
completely the equality case.

######  Lemma 2.2.

The curvature gap $\delta$ of any flat sphere satisfies
$\delta\leq\frac{1}{3}$. The equality holds if and only if all of conical
angles belong to $2\pi+\frac{4\pi}{3}\mathbb{Z}$.

Equivalently, the equality case for the curvature gap corresponds to flat
metrics of finite area induced by meromorphic cubic differentials with
singularities of even order.

###### Proof.

We assume by contradiction that there is a sequence of numbers
$k_{1},\dots,k_{n}\in]-\infty,1[$ such that:

  * •

$\sum\limits_{i=1}^{n}k_{i}=2$;

  * •

no partial sum belongs to the interval $]\frac{2}{3},\frac{4}{3}[$;

Without loss of generality, we assume this sequence has a minimal number $n$
of elements among sequences with this property (we obviously have $n\geq 3$).

It follows from the hypothesis on partial sums that
$k_{1},\dots,k_{n}\in]-\infty,\frac{2}{3}]$. Now there are two cases depending
on the value of $k_{1}+k_{2}$. If $k_{1}+k_{2}<1$, we replace $k_{1},k_{2}$ by
$k_{1}+k_{2}$ in the sequence and get a sequence with a smaller $n$ satisfying
desired properties. On the opposite, if $k_{1}+k_{2}\geq 1$, we have
$k_{1}+k_{2}\in[1,\frac{4}{3}]$ and we contradict the hypothesis on partial
sums.

If a flat surface has a curvature gap $\delta=\frac{1}{3}$, no partial sum of
its discrete curvatures $k_{1},\dots,k_{n}$ can belong to the open interval
$]\frac{2}{3},\frac{4}{3}[$. Positive discrete curvatures are thus contained
in the interval $]0,\frac{2}{3}]$. Since their total sum is at least $2$,
there is a partial sum at least $\frac{4}{3}$. It forces that two partial sum
of positive curvatures are $\frac{2}{3}$ and $\frac{4}{3}$ respectively. It
follows that there is a curvature equal to $\frac{2}{3}$. If there is a
positive curvature less than $\frac{2}{3}$, the sum of this curvature with
$\frac{2}{3}$ is in the open interval $]\frac{2}{3},\frac{4}{3}[$, which is
impossible. Thus, every positive number among $k_{1},\dots,k_{n}$ is equal to
$\frac{2}{3}$.

Without loss of generality, we assume that $k_{1}=\dots=k_{s}=\frac{2}{3}$ and
$k_{t}\leq 0$ for any $t>s$. Then, for any such $k_{t}$,
$k_{t}+\sum\limits_{i=1}^{s}k_{i}\geq 2$, unless
$k_{t}\in\frac{2}{3}\mathbb{Z}$, at least one of the partial sums
$k_{t}+\sum\limits_{i=1}^{l}k_{i}$ for $l\leq s$ belongs to the open interval
$]\frac{2}{3},\frac{4}{3}[$. This would contradict $\delta=\frac{1}{3}$. It
follow that every discrete curvature $k_{i}$ is an integer multiple of
$\frac{2}{3}$.

The corresponding flat metrics have conical angles in
$2\pi+\frac{4\pi}{3}\mathbb{Z}$. For any developing map $f$ of the metric,
cubic differential $(df)^{\otimes 3}$ is unaffected by the monodromy around
conical singularities (formed by translations and rotations of order $3$).
Conversely, zeroes and poles of order $k\geq-2$ of a cubic differential
$\omega$ correspond to conical singularities of angle $\frac{(k+3)2\pi}{3}$ in
the metric induced by differential $|\omega^{1/3}|$ (see [2] for the
background on differentials of higher orders). ∎

##  3\. Flat annuli

Just like the geometry of a translation surface is largely determined by
shapes of its flat cylinders, flat annuli play an analogous role for flat
surfaces.

######  Definition 3.1.

For any angle $\theta\in]0,2\pi]$ and radii $0\leq R<R^{\prime}$, we define a
flat annulus $\mathcal{A}_{R,R^{\prime},\theta}$ by identifying the two radial
sides of domain $\\{{z\in\mathbb{C}\leavevmode\nobreak\ |\leavevmode\nobreak\
R\leq|z|\leq R^{\prime}\leavevmode\nobreak\ ;\leavevmode\nobreak\ 0\leq
arg(z)\leq\theta\\}}$ by a rotation of the angle $\theta$.

We refer to $\theta$ as the apex angle of the annulus. The definition extends
for any $\theta>0$ by a covering construction.

The exponential map provides a holomorphic parametrization of the flat annulus
$\mathcal{A}_{R,R^{\prime},\theta}$ by the flat cylinder obtained from a
rectangle $\\{{z\in\mathbb{C}\leavevmode\nobreak\ |\leavevmode\nobreak\ 0\leq
Re(z)\leq ln(R^{\prime}/R)\leavevmode\nobreak\ ;\leavevmode\nobreak\ 0\leq
Im(z)\leq\theta\\}}$. The conformal modulus of the flat annulus
$\mathcal{A}_{R,R^{\prime},\theta}$ is thus $\frac{ln(R^{\prime}/R)}{\theta}$.

Identifying the center of the flat annulus with $0\in\mathbb{C}$, the polar
coordinate of the complex plane provides a pair of orthogonal foliations on
$\mathcal{A}_{R,R^{\prime},\theta}$:

  * •

the radial foliation, whose leaves are trajectories contained in lines
incident to the center of the annulus;

  * •

the orthoradial foliation, whose leaves are circular arcs orthogonal to radial
leaves.

In particular, the curvature radius of orthoradial leaves varies between $R$
and $R^{\prime}$. We will refer to boundaries of orthoradial leaves of radii
$R$ and $R^{\prime}$ respectively as the inner boundary arc and the outer
boundary arc. If $R=0$, the inner boundary arc is reduced to a conical
singularity of the angle $\theta$.

###  3.1. Trajectories in a flat annulus

In any annulus $\mathcal{A}_{R,R^{\prime},\theta}$, there is an important
difference between trajectories starting from the inner boundary arc and
trajectories starting from the outer boundary arc. We obtain this description
by looking at the lift of the trajectory in the universal cover of the flat
annulus.

In any trajectory $t$ starting from the inner boundary arc, the radius
(determining a leaf in the orthogonal foliation) along the trajectory is
strictly increasing from $R$ to $R^{\prime}$. It follows that it is simple and
leaves the annulus through the outer boundary arc.

For a trajectory $t$ starting from the outer boundary arc, we denote by $A$
the starting point of $t$ and $\alpha$ the angle of $t$ with the radial leaf
incident to $A$. There are two different regimes (see Figure 1):

Figure 1. Two trajectories in the universal cover of a flat annulus.

  * •

if $\alpha\leq\arcsin(R^{\prime}/R)$, the radius is strictly decreasing from
$R^{\prime}$ to $R$ and trajectory $t$ eventually leaves the annulus through
the inner boundary arc. The trajectory is automatically simple.

  * •

if $\alpha>\arcsin(R^{\prime}/R)$, the radius is strictly decreasing from
$R^{\prime}$ to $R^{\prime}\sin(\alpha)$ and then strictly increasing from
$R^{\prime}\sin(\alpha)$ to $R^{\prime}$. The (possibly self-intersecting)
trajectory eventually leaves the annulus through the outer boundary.

In the second case, the trajectory is a chord of the outer circle whose
central angle is $\pi-2\alpha$. We observe that all the self-intersections of
$t$ belong to two radial leaves (see Figure 2):

  1. (1)

the radial leaf $\mathcal{L}$ containing the unique point where $t$ is tangent
to the orthoradial foliation and realizes the minimal radius
$R^{\prime}\sin(\alpha)$;

  2. (2)

the radial leaf obtained from $\mathcal{L}$ by a rotation of angle
$\frac{\theta}{2}$.

A direct counting then proves that the number of self-intersections in
trajectory $t$ is $\lfloor\frac{\pi-2\alpha}{\theta}\rfloor$.

Figure 2. The thin and thick red radial leaves are the copies of the leaves
where self-intersections of $t$ belong to.

###  3.2. Monogonal trajectories

In a flat sphere, a monogonal trajectory is a simple trajectory with the same
endpoint and avoids any conical singularity. We call the endpoint the turning
point. Notice that the trajectory divides the angle at the turning point into
two sub-angles. We call the smaller sub-angle the interior angle.

The curvature gap provides constraints on interior angles of monogonal
trajectories.

######  Lemma 3.2.

In a flat sphere $X$ with a curvature gap $\delta$, for any monogonal
trajectory $\gamma$, the interior angle $\alpha$ of loop $\gamma$ satisfies
$\alpha\leq\pi-2\pi\delta$.

###### Proof.

Any closed topological loop $\gamma$ decomposes $X$ into connected components
and the set of conical singularities into two subsets. If $\gamma$ is a
polygonal path with an interior angle, Gauss-Bonnet formula implies that
$\pi-\alpha$ equals to $2\pi$ times the total curvature encompassed by the
loop. By the definition of the curvature gap, our inequality follows. ∎

A monogonal trajectory $\gamma$ is characterized by a pair $(\alpha,L)$ where
$\alpha\in]0,\pi[$ is the interior angle while $L$ is the length of the
segment.

The holonomy of the flat metric along the loop is a rotation of angle
$\pi-\alpha$. Each point of $\gamma$ has a well-defined radius with respect
with the fictive center of this rotation. A direct trigonometric computation
proves that the maximal radius $\frac{L}{2\cos(\alpha/2)}$ is realized at the
turning point while the minimal radius $\frac{L\tan(\alpha/2)}{2}$ is realized
at the midpoint of the unique segment of $\gamma$.

Each monogonal trajectory $\gamma$ can be embedded in a unique way in a
$1$-parameter family of disjoint monogonal trajectory
$(\gamma_{l})_{L<l<L^{\prime}}$ such that:

  * •

the interior angle of each trajectory of the family are the same;

  * •

turning points of monogonal trajectories describe a straight line
$\mathcal{L}$ in the flat surface as $l$ changes;

  * •

$\mathcal{L}$ is the bisector of the interior angle for each trajectory
$\gamma_{l}$.

Provided $\frac{L^{\prime}}{L}>\frac{1}{\sin(\alpha/2)}$, notice that the flat
surface which is swept by a family $(\gamma_{l})_{L<l<L^{\prime}}$ contains an
annulus $\mathcal{A}_{\theta,R,R^{\prime}}$ (see Figure 3) where:

  * •

$\theta=\pi-\alpha$;

  * •

$R=\frac{L}{2\cos(\alpha/2)}$;

  * •

$R^{\prime}=\frac{L^{\prime}\tan(\alpha/2)}{2}$.

Figure 3. A flat annulus is contained in a family of monogonal trajectories.

###  3.3. Counting saddle connections in a flat annulus

This section is dedicated to explaining why the curvature gap is necessary for
our estimates in Theorem 1.2 and Theorem 1.3.

For $0\leq R<R^{\prime}$ and $0<\theta<\pi$, we consider a flat annulus
$\mathcal{A}_{R,R^{\prime},\theta}$ embedded in a flat sphere such that each
boundary arc contains exactly one singularity. We will also assume that these
singularities $A,B$ belong to the same radial leaf.

######  Proposition 3.3.

Flat annulus $\mathcal{A}_{R,R^{\prime},\theta}$ contains at least
$\arccos{(\frac{R}{R^{\prime}})}\cdot\frac{1}{\theta}$ simple saddle
connections.

For a flat sphere $X$ containing a flat annulus
$\mathcal{A}_{R,R^{\prime},\theta}$, Lemma 3.2 proves that apex angle $\theta$
satisfies a lower bound $\theta\geq 2\pi\delta$ where $\delta$ is the
curvature gap $\delta$ of $X$. When the apex angle $\theta$ goes to zero, the
curvature gap of the surface goes to zero and the number and the length of
simple saddle connections tend to infinity. This indicates why the curvature
gap $\delta$ is necessary in the upper bound of Theorem 1.2.

Taking the limit $\theta\rightarrow 0$ while bounding the conformal modulus of
the flat annulus, we obtain in the limit a flat cylinder containing infinitely
many saddle connections with arbitrarily large lengths. In a flat sphere, this
can only happen if $\delta=0$.

###### Proof of Proposition 3.3.

We develop counterclockwise the flat annulus and denote by
$A_{0},A_{1},\dots,A_{k}$ the copies of $A$ as in Figure 4.

We define $I$ as the tangency point for a line $IB$ incident to $B$ and
tangent to the circle of radius $r$. Defining $\beta$ as $\angle IOB$, $I$
belongs to the circular arc between two successive copies $A_{m}$ and
$A_{m+1}$ of $A$, see Figure 4.

Since $\cos(\beta)=\frac{R}{R^{\prime}}$, we have
$m\theta\leq\arccos(\frac{R}{R^{\prime}})\leq(m+1)\theta$. It follows that
segments $[BA_{0}],\dots,[BA_{m}]$ form a family of
$\arccos(\frac{R}{R^{\prime}})\cdot\frac{1}{\theta}$ simple saddle
connections.

Figure 4. An unfolding of the almost cylinder and the sequence of saddle
connections $[BA_{0}],\dots,[BA_{m}]$.

∎

##  4\. Geometry of immersed disks

###  4.1. Locally isometric immersions of disks

In the study of a flat surface, a crucial geometric quantity is the size of
the largest open disk we can immerse or embed in the surface without hitting
the conical singularities.

######  Proposition 4.1.

If there is a locally isometric immersion $f$ of an open disk of radius
$\lambda>\frac{1}{\sqrt{\pi}}$ in a flat sphere $X$ of a unit area, then $X$
contains an embedded flat annulus $\mathcal{A}_{R,R^{\prime},\theta}$ where:

  * •

the apex angle $\theta$ satisfies $\theta<\frac{\pi}{4\pi\lambda^{2}-4}$;

  * •

the conformal modulus $\mu$ satisfies $\mu>\sqrt{\pi\lambda^{2}-1}$.

###### Proof.

We identify the domain of immersion $f$ with the centered disk $\mathcal{D}$
of radius $\lambda$ in the complex plane and set $M=f(0)$. Since $f$ is
locally injective, $M$ is a center of a maximal embedded disk
$\mathcal{D}^{o}$ in $X$. The area of $\mathcal{D}^{o}$ is bounded by the area
of $X$ so the radius $\rho$ of $\mathcal{D}^{o}$ satisfies
$\rho<\frac{1}{\sqrt{\pi}}$. In particular, we have $\lambda>\rho$ and
$\mathcal{D}^{o}$ is contained in $\mathcal{D}$.

We deduce from the maximality hypothesis on $\mathcal{D}^{o}$ that two points
$A,B$ of $\partial\mathcal{D}^{o}$ have the same image under the immersion.
Without loss of generality, we suppose that the mediatrix of the segment
$[A,B]$ contains the horizontal diameter of disk $\mathcal{D}$. We locate
points $A,B$ with corresponding complex numbers $\rho e^{i\alpha}$ and $\rho
e^{i\alpha}$ respectively for $\alpha\in]0,\frac{\pi}{2}[$ see Figure 5.

Figure 5. Delaunay disk $\mathcal{D}$, embedded disk $\mathcal{D}^{o}$,
monogonal trajectory $[AB]$ and the flat annulus in grey.

A neighborhood of $B$ in the tangent line $T_{B}$ of $\partial\mathcal{D}^{o}$
of $B$ has the image under $f$ as some segment containing $A$. Since
$\mathcal{D}^{o}$ is the maximal embedded disk centered on $M$, this segment
has also to be contained in the tangent line $T_{A}$ to
$\partial\mathcal{D}^{o}$. It follows that segment $[AB]$ belongs to a family
of vertical monogonal trajectories (see Section 3.2) with turning points
located at these identified tangent lines. This family of curves can be
continued on $X$ until reaching a conical singularity. Since $f(\mathcal{D})$
cannot contain any conical singularity, it follows that the intersection point
$N$ of these two tangent lines is outside $\mathcal{D}$ (possibly at
infinity), see Figure 5. Identify point $N$ with the real number $x$ of the
complex plane, we have $x\geq\lambda$.

Each of the two identified points between tangent lines $T_{A},T_{B}$ is
connected by a circular arc of the center $N$ contained in disk $\mathcal{D}$.
We denote by $R<R^{\prime}$ the radii of these arcs. The images under $f$ of
these two circular arcs bound a flat annulus
$\mathcal{A}_{R,R^{\prime},\theta}$ where apex angle satisfies
$\theta=\pi-2\alpha$.

The area $A=\theta(R^{\prime}+R)(R^{\prime}-R)$ of this annulus is bounded by
the total area of the surface. Since we have
$R^{\prime}-R=2\sqrt{\lambda^{2}-\rho^{2}}$ and $\rho^{2}>\frac{1}{\pi}$, we
deduce the inequality $A>4\theta(\lambda^{2}-\frac{1}{\pi})$. It follows from
$A<1$ that $\theta<\frac{\pi}{4\pi\lambda^{2}-4}$.

It remains to give a lower bound on the conformal modulus of
$\mathcal{A}_{R,R^{\prime},\theta}$. In fact, the modulus is
$\frac{ln(R^{\prime}/R)}{\theta}$. We know $\theta=\pi-2\alpha$. Since
$R^{\prime}-R=2\sqrt{\lambda^{2}-\rho^{2}}$ and
$R^{\prime}+R=2\rho\tan{\alpha}$, we have
$R^{\prime}=\rho\tan{\alpha}+\sqrt{\lambda^{2}-\rho^{2}}$ and
$R=\rho\tan{\alpha}-\sqrt{\lambda^{2}-\rho^{2}}$. Hence, the conformal modulus
$\mu$ of $\mathcal{A}_{R,R^{\prime},\theta}$ is given by:

$\mu=\frac{\ln\left({\frac{\rho\tan{\alpha}+\sqrt{\lambda^{2}-\rho^{2}}}{\rho\tan{\alpha}-\sqrt{\lambda^{2}-\rho^{2}}}}\right)}{\pi-2\alpha}.$

According to the inequality $\ln(1+x)\geq\frac{x}{1+x}$ for $x>-1$, we know
that:

$\frac{\ln\left({\frac{\rho\tan{\alpha}+\sqrt{\lambda^{2}-\rho^{2}}}{\rho\tan{\alpha}-\sqrt{\lambda^{2}-\rho^{2}}}}\right)}{\pi-2\alpha}\geq\frac{2\sqrt{\lambda^{2}-\rho^{2}}}{\rho\tan{\alpha}-\sqrt{\lambda^{2}-\rho^{2}}}\cdot\frac{1}{\pi-2\alpha}.$

As $\alpha$ increases, the right side of the inequality decreases. In the
limit $\alpha\longrightarrow\frac{\pi}{2}$, the flat annulus becomes a flat
cylinder and we obtain:

$\frac{2\sqrt{\lambda^{2}-\rho^{2}}}{\rho\tan{\alpha}-\sqrt{\lambda^{2}-\rho^{2}}}\cdot\frac{1}{\pi-2\alpha}\geq\frac{\sqrt{\lambda^{2}-\rho^{2}}}{\rho}.$

It follows then from inequality $\rho<\frac{1}{\sqrt{\pi}}$ that the conformal
modulus $\mu$ of $\mathcal{A}_{R,R^{\prime},\theta}$ is strictly larger than
$\sqrt{\pi\lambda^{2}-1}$. ∎

######  Remark 4.2.

An interesting geometric result due to Masur and Smillie (Corollary 5.5 in
[5]) proves that a translation surface of large diameter contains a long flat
cylinder. Proposition 4.1 provides the analog result for flat spheres (and in
fact flat surfaces of arbitrary genus since the proof relies on no topological
assumption), see [7] for discussion of flat spheres near the boundary of the
moduli space.

###  4.2. Delaunay triangulations

In any flat sphere $X$ (and more generally any flat surface with conical
singularities), the Voronoi cell of a singularity $M$ is the polygonal domain
in $X$ consisting of points that are closer (with respect to the flat metric)
to $M$ than to any other conical singularity of $X$. This defines the Voronoi
tessellation of $X$.

The Delaunay polygonation is a construction dual to the Voronoi tessellation.
We consider locally isometric immersions of disks into $X$. When at least
three points of the boundary circle are mapped to conical singularities of
$X$, these disks are called Delaunay disks and the immersion of the convex
hull of these singularities into $X$ is called a Delaunay polygon. They form a
decomposition of $X$ dual to the Voronoi tessellation, see Proposition 3.1 of
[7] for details on these constructions.

If more than three conical singularities of $X$ are cocyclic, some Delaunay
polygons may be convex polygons with more than three sides. A Delaunay
triangulation is any subdivision of the Delaunay polygonation into triangles.
We refer to edges of such a triangulation as Delaunay edges.

The following lemma uses the curvature gap to bound the length of Delaunay
edges on any flat sphere.

######  Lemma 4.3.

In a flat sphere $X$ of a unit area with $n$ conical singularities and a
curvature gap $\delta>0$, the length $L$ of any edge of a Delaunay
triangulation of $X$ satisfies $L^{2}<\frac{4}{\pi}+\frac{1}{2\pi\delta}$.

###### Proof.

We assume by contradiction that the length $\lambda$ of some Delaunay edge
satisfies $L^{2}\geq\frac{4}{\pi}+\frac{1}{2\pi\delta}$. It follows that it is
contained in the image of a Delaunay disk $\mathcal{D}$ of radius
$\lambda\geq\frac{L}{2}$ by a locally isometric immersion $f$.

Radius $\lambda$ is by hypothesis strictly larger than $\frac{1}{\sqrt{\pi}}$.
It follows then from Proposition 4.1 that the image of $\mathcal{D}$ contains
a flat annulus of apex angle $\theta<\frac{\pi}{4\pi\lambda^{2}-4}$.
Orthoradial leaves of this flat annulus (see Section 3) decompose $X$ into two
connected components whose total curvatures are respectively
$\frac{2\pi-\theta}{2\pi}$ and $\frac{2\pi+\theta}{2\pi}$. It follows then
from the definition of the curvature gap that we have $\theta\geq 2\pi\delta$
and thus we get $\lambda^{2}<\frac{1}{\pi}+\frac{1}{8\pi\delta}$. The bound on
$L$ follows. This contradicts our assumption. ∎

Locally isometric immersions of disks provide also our main tool to control
the complexity of simple trajectories in a flat sphere.

######  Lemma 4.4.

In a flat sphere $X$, we consider a Delaunay triangle $\mathcal{T}$ and its
Delaunay disk locally isometric immersion $f:\mathcal{D}\rightarrow X$.

Assuming that the curvature gap $\delta$ of $X$ satisfies $\delta>0$, for any
simple saddle connection $t$, the intersection $f^{-1}(t)\cap\mathcal{T}$
consists of at most $\frac{5}{2\delta}$ chords of disk $\mathcal{D}$.

###### Proof.

Since $t$ is a saddle connection, its endpoints do not belong to the image of
the interior of a Delaunay disk. It follows that $f^{-1}(t)$ is formed by
chords of the disk $\mathcal{D}$. Since $t$ is simple, these chords do not
intersect each other. We denote by $\mathcal{C}h$ the (possibly infinite)
subset of these chords crossing the Delaunay triangle $\mathcal{T}$. Let $m$
be the number of chords in $\mathcal{C}h$.

Cutting along chords of $\mathcal{C}h$, we decompose the triangle
$\mathcal{T}$ into several connected components. The dual graph of the
decomposition is a tree of $m$ edges with at most three branches (one for each
corner of the triangle). We deduce that the dual graph contains a sub-path
formed by $k$ edges where $k\geq\frac{2m}{3}$.

We denote by $\mathcal{C}h^{0}$ the union of chords corresponding to edges of
this subgraph. The linear graph provides a natural linear order of these
chords $c_{1},\dots,c_{k}$. We denote by $\theta_{1},\dots,\theta_{k}$
directions of tangent vectors of these chords. There is a permutation
$\sigma\in\mathfrak{S}_{k}$ such that $c_{\sigma(1)},\dots,c_{\sigma(k)}$ is
the order on chords induced by the orientation of $t$. For each chord $c_{j}$,
we choose a point $A_{j}\in c_{j}\cap\mathcal{T}$.

For any $1\leq j\leq k-1$, we denote by $I_{j}$ the domain in $\mathcal{D}$
bounded by $c_{j}$ and $c_{j+1}$. We are going to estimate the total arc-
length $\beta_{i}$ of $\partial\mathcal{D}\cap I_{j}$ for each $j$.

We first consider the case where two consecutive chords $c_{j}$ and $c_{j+1}$
have the same orientation (for the orientation of the trajectory $t$). Without
loss of generality, we assume that $\sigma(j)<\sigma(j+1)$. Inside
$\mathcal{T}$, we draw an oriented segment $\gamma$ from $A_{j+1}$ to $A_{j}$.
The path $\gamma$ and the portion $[A_{j}A_{j+1}]$ of the simple trajectory
$t$ form a closed polygonal loop with no self-intersection. We deduce then
from Lemma 3.2 that directions of $c_{j}$ and $c_{j+1}$ cannot form an angle
smaller than $2\pi\delta$. It follows then from the inscribed angle theorem
that $\beta_{i}\geq 4\pi\delta$.

Now we deal with the case where three consecutive chords
$c_{j},c_{j+1},c_{j+2}$ have alternate orientations, see Figure 6. Up to
reversing the global orientation of trajectory $t$, there are just two cases
to consider:

  1. (1)

$\sigma(j)<\sigma(j+2)<\sigma(j+1)$;

  2. (2)

$\sigma(j)<\sigma(j+1)<\sigma(j+2)$.

Figure 6. Delaunay disk $\mathcal{D}$, Delaunay triangle $\mathcal{T}$ and
three consecutive chords $c_{j},c_{j+1},c_{j+2}$ with alternate orientations.

We draw an oriented segment $\gamma$ from $A_{j+2}$ to $A_{j}$ in the triangle
$\mathcal{T}$. It intersects the chord $c_{j+1}$ in some point $M$. In the
case (1), the portion $[A_{j}A_{j+2}]$ of the simple trajectory $t$ does not
contain chord $c_{j+1}$ and the loop formed by the path $\gamma$ and
$[A_{j}A_{j+2}]$ is a closed polygonal loop with no self-intersection. For the
same reasons as previously, we get $\beta_{j}+\beta_{j+1}\geq 4\pi\delta$.

In case (2), the path $\gamma$ and the portion $[A_{j}A_{j+2}]$ of simple
trajectory $t$ form a eight-shaped curve in flat surface $X$. It can be
described as follows:

  * •

The double point of the eight-shaped curve is $M$.

  * •

A first arc is formed by oriented portion $[MA_{j}]$ of segment $\gamma$ and
oriented portion $[A_{j}M]$ of trajectory $t$. It encompasses negatively a
total curvature $\kappa_{1}$.

  * •

A second arc is formed by oriented portion $[A_{j+2}M]$ of segment $\gamma$
and oriented portion $[MA_{j+2}]$ of trajectory $t$. It encompasses positively
a total curvature $\kappa_{2}$.

Each interior angle of these two loops is strictly smaller than $2\pi$. We
deduce from the Gauss-Bonnet formula that $0<\kappa_{1},\kappa_{2}<1$.
Besides, it follows from the definition of the curvature gap that in fact, we
have $\kappa_{1},\kappa_{2}\leq 1-\delta$.

If $|\theta_{j+2}-\theta_{j}|\geq 2\pi\delta$, we prove as in previous cases
that $\beta_{j}+\beta_{j+1}\geq 4\pi\delta$. We will thus assume that
$|\theta_{j+2}-\theta_{j}|<2\pi\delta$. The only case where we can have
$\beta_{j}+\beta_{j+1}<4\pi\delta$ implies that
$|\theta_{j}-\theta_{j+1}-\pi|+|\theta_{j+2}-\theta_{j+1}-\pi|<2\pi\delta$. We
deduce that $|\kappa_{1}-\frac{1}{2}|+|\kappa_{2}-\frac{1}{2}|<\delta$. It
follows that $|\kappa_{1}+\kappa_{2}-1|<\delta$ and we obtain a contradiction.
Consequently, we have $\beta_{j}+\beta_{j+1}\geq 4\pi\delta$ in any case.
Observe that in this last case, applying the Gauss-Bonnet formula to the
eight-shaped loop, we obtain that $|\kappa_{1}-\kappa_{2}|<\delta$ (indeed, a
portion of this curvature is encompassed positively while the other is
encompassed negatively).

If $k\geq 3$, then in the worst case, no pair of consecutive chords of
$\mathcal{C}h^{0}$ have the same orientation and we obtain that
$\sum\limits_{j=1}^{k-1}\beta_{j}$ is no less than $2\pi(k-1)\delta$ when $k$
is odd and $2\pi(k-2)\delta$ when $k$ is even. It follows that $k\leq
2+\frac{1}{\delta}$ and thus $m\leq\frac{3+6\delta}{2\delta}$. Since
$\delta\leq\frac{1}{3}$ (see Lemma 2.2), we deduce that
$m\leq\frac{5}{2\delta}$.

If $k\leq 2$, then we have $m\leq 3$ and $m\leq\frac{5}{2\delta}$ holds also
in this case. ∎

######  Remark 4.5.

Since it is a local construction, Lemma 4.4 holds in fact in a much more
general settings. We just have to assume that the disk immersion preserves a
complex affine structure. Proposition 5.2 and Theorem 1.2 will also extend to
complex affine surfaces of genus zero that admit a Delaunay triangulation.

###  4.3. Essentially sharp upper bound on diameter

We deduce from Lemma 4.3 an upper bound on the diameter of a flat sphere.

###### Proof of Corollary 1.4.

For any pair of points $M,N$ in the flat sphere, we are going to prove that
the distance between $M$ and $N$ induced by the flat metric is at most
$(n+1)(\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}})$.

Unless $M$ and $N$ are already conical singularities, we mark them (the number
of singularities is then at most $n+2$) and consider a Delaunay triangulation
compatible with this new set of singularities.

The Delaunay edges form a metric graph embedded in $X$. The shortest path
between $M$ and $N$ in the graph is formed by at most $n+1$ edges. Lemma 4.3
proves that each Delaunay edge is of length strictly less than
$\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}}$. It follows that the
distance between $M$ and $N$ is smaller than
$(n+1)(\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}})$. A bound on the
diameter follows. ∎

Our upper bound on the diameter is of order $\frac{1}{\sqrt{\delta}}$. In
fact, this upper bound is essentially sharp which means that there is a
surface whose diameter has a lower bound of order $\frac{1}{\sqrt{\delta}}$.
We will construct such a surface in the following example. It also implies
that an upper bound on the diameter has to depend on the number of
singularities.

######  Example 4.6.

Consider a flat sphere $X$ constructed as in Figure 7. $l^{\prime}$ is an
arbitrary positive number and $l\in(0,l^{\prime})$. $X$ has $m$ singularities
of curvatures $\epsilon$, $m-1$ singularities of curvatures $-\epsilon$ and
two singularities of curvatures $1-\frac{\epsilon}{2}$. Notice that when
$\epsilon<\frac{2}{2m+1}$, the curvature gap $\delta$ of $X_{\epsilon}$ is
$\frac{\epsilon}{2}$.

Figure 7. Edges of the polygon are glued with respect to numbers and arrows.
Curvatures of $A$ and $B$ are both $1-\frac{\epsilon}{2}$, black vertices are
singularities of the curvature $\epsilon$ and grey vertices are singularities
of the curvature $-\epsilon$.

Notice that the distance between two singularities of curvatures
$1-\frac{\epsilon}{2}$ is realized by the diagonal between them in the
polygon. The length of this diagonal is $2l^{\prime}+2(m-1)l>2ml$. The area of
the surface is $2[l^{\prime
2}+(m-1)(2ll^{\prime}-l^{2})]\tan\frac{\pi\epsilon}{2}<4ml^{\prime
2}\epsilon$.

Hence, the normalized diameter of $X$ is at least
$\frac{l}{l^{\prime}}\sqrt{\frac{m}{\epsilon}}=\frac{l}{l^{\prime}}\sqrt{\frac{m}{2\delta}}$.
In particular, this implies that we cannot hope a uniform upper bound that
does not depend on the number of singularities.

##  5\. Upper bounds

###  5.1. Bounding combinatorial lengths of trajectories

######  Definition 5.1.

In a flat sphere endowed with a fixed Delaunay triangulation, for any non-
periodic trajectory $t$ in $X$, we define the combinatorial length of $t$ as
the number of segments of $t$ in the complement of $X$ of the union of
Delaunay edges.

In the following proposition, we use the curvature gap to give an upper bound
on the combinatorial length of any simple trajectory.

######  Proposition 5.2.

We consider a flat sphere $X$ with $n$ conical singularities, a curvature gap
$\delta>0$ and a fixed Delaunay triangulation. For any simple trajectory $t$
in $X$, unless $t$ is contained in a Delaunay edge, the combinatorial length
of $t$ is at most $\frac{5n}{\delta}$.

###### Proof.

We first consider the case where $t$ is a saddle connection. Any Delaunay
triangle $\mathcal{T}$ is contained in the image of some locally isometric
immersion $f$ of a disk $\mathcal{D}$. Following Lemma 4.4, the pullback of
$t$ by this immersion is formed by at most $\frac{5}{2\delta}$ chords crossing
the triangle.

Since there are $2n-4$ triangles in the Delaunay triangulation, the
intersection of $t$ with the complement of Delaunay edges is formed by at most
$(2n-4)\frac{5}{2\delta}$ distinct segments. We deduce an upper bound on the
number of intersections with Delaunay edges.

If $t$ is not a saddle connection, we can mark its endpoints to create conical
singularities of angle $2\pi$. This does not change the value of $\delta$ but
may increase the number of triangles in the Delaunay triangulation by $2$. The
upper bound obtained is $\frac{5n}{\delta}$ instead.

Note that the trajectory cannot be periodic because it would cut out the flat
sphere into two components and force the curvature gap to be zero. ∎

We next prove an upper bound on the number of simple saddle connections using
Proposition 5.2.

Kneser introduced in [4] normal coordinates to represent curves. We define a
similar coordinate in our settings. A simple path refers to a topological path
which connects conical singularities and has no self-intersections or conical
singularities in the interior. Given a triangulation $T$, we call a simple
path normal with respect to $T$ if it intersects with edges of $T$
transversely except at its endpoints and it enters and leaves a triangle of
$T$ via different edges. In particular, if a simple saddle connection is not
an edge of a Delaunay triangulation, it is normal with respect to the Delaunay
triangulation.

Denote by $p$ a normal simple path. For every edge $e$ of $T$, we define a
non-negative integer to be the number of intersections of $p$ with the
interior of the edge $e$, denoted by $\gamma(e)$. Since there are $3(n-2)$
many edges in $T$, we obtain a vector of $3(n-2)$ non-negative integers in
$\mathbb{R}^{3(n-2)}$ and we call it the normal coordinate of $p$.

######  Lemma 5.3.

Given a triangulation $T$, a normal coordinate determines a normal simple path
with respect to $T$ up to isotopy.

###### Proof.

Notice that triangles of $T$ cut the normal path $p$ into sub-paths
$p_{1},\dots,p_{k}$. The order of these sub-paths is according to the
orientation of $p$. $p_{1}$ and $p_{2}$ are special because they start or end
in a vertex of $T$.

For a triangle of $T$ that does not contain $p_{1}$ or $p_{k}$, we denote
edges of the triangle by $e_{1},e_{2},e_{3}$. Notice that
$p(e_{1})+p(e_{2})+p(e_{3})$ is even and $p(e_{i})+p(e_{j})\geq p(e_{k})$ for
$\\{i,j,k\\}=\\{1,2,3\\}$. As normal coordinates in [4], for fixed
$p(e_{1}),p(e_{2}),p(e_{3})$, there is only one way to pair intersections on
different edges by arcs in the triangle such that these arcs do not intersect
in the triangle (see Figure 8).

Figure 8. The $\\{3,4,5\\}$ determines disjoint arcs in the triangle up to
isotopy.

For $p_{1}$ and $p_{k}$, there are two possibilities. If they are in two
different triangles, for each of triangles, we can find a way to name edges of
triangles by $e_{1},e_{2},e_{3}$ such that $p(e_{3})=p(e_{1})+p(e_{2})+1$. If
they are in the same triangle, we can name edges of the triangle by
$e_{1},e_{2},e_{3}$ such that $p(e_{3})=p(e_{1})+p(e_{2})+2$. In either case,
once $p(e_{1}),p(e_{2}),p(e_{3})$ are fixed, there is only one way to pair
intersections on different edges by arcs such that these arcs do not intersect
in the triangle (see Figure 9). Hence, a normal coordinate corresponds to an
unique way to reconstruct a normal path up to isotopy.

Figure 9. The left figure shows the case when a triangle contains only the
first sub-path. The right figure shows the case when a triangle contains the
first and the last sub-paths.

∎

######  Corollary 5.4.

In a flat sphere with $n$ conical singularities and a curvature gap of
$\delta>0$, the number of simple saddle connections is at most
$\frac{(\frac{5n}{\delta}+3n-7)^{3n-6}}{(3n-7)!}+3n-6$.

###### Proof.

We shall prove an upper bound on the number of saddle connections crossing
exactly $k$ Delaunay edges. Denote by $\gamma$ such a saddle connection. If
$\gamma$ is an edge of the Delaunay triangulation, there are $3n-6$ many
choices. If $\gamma$ is not an edge of the Delaunay triangulation, it is
normal with respect to the Delaunay triangulation. It follows that the normal
coordinate of $\gamma$ is well-defined. Since $\gamma$ crosses exactly $k$
Delaunay edges, the sum of the integer $\gamma(e)$ over all Delaunay edges is
equal to $k$. According to Lemma 5.3, the number of saddle connections
crossing exactly $k$ Delaunay edges is the same as the number of weak
compositions of the integer $k$, which is
$\left(\begin{array}[]{l}k+3(n-2)-1\\\
3(n-2)-1\end{array}\right)<\frac{(k+3n-7)^{3n-7}}{(3n-7)!}$.

Summing all cases up, the total number of saddle connections crossing at most
$k$ saddle connections is at most $(3n-6)+\sum\limits_{1\leq i\leq
k}\frac{(k+3n-7)^{3n-7}}{(3n-7)!}<3n-6+\frac{(k+3n-7)^{3n-6}}{(3n-7)!}$. Since
Proposition 5.2 tells us that $k<\frac{5n}{\delta}$, we obtain the upper bound
$\frac{(\frac{5n}{\delta}+3n-7)^{3n-6}}{(3n-7)!}+3n-6$. ∎

###  5.2. Trajectories with self-intersections

In order to bound the combinatorial length of nonsimple trajectories, we
combine geometric estimates with purely topological bounds on isotopically
disjoint loops. We start by giving a rigorous definition of the number of
self-intersections of a trajectory (see [3] for the notion of self-
intersections for geodesics on hyperbolic surfaces).

######  Definition 5.5.

At any self-intersection point $p$ of a trajectory $t$ in a flat sphere $X$,
we say that two tangent vectors to $t$ (for the orientation of $t$) form a
transverse pair if they span the tangent plane at $p$. The total number
$|t\cap t|$ of transverse pairs (summed on the set of self-intersection points
of $t$) is the self-intersection number of $t$.

The following topological lemma follows directly from the pairs-of-pants
decomposition.

######  Lemma 5.6.

In a topological sphere with $n\geq 3$ punctures, the maximal number of
isotopically disjoint homotopically nontrivial loops is $2n-3$.

A monogonal trajectory (see Section 3.2) $\gamma$ decomposes a flat sphere $X$
into two connected components:

  * •

the inner component $X_{\gamma}^{-}$ where the total curvature is strictly
lower than $1$;

  * •

the outer component $X_{\gamma}^{+}$ where the total curvature is strictly
higher than $1$.

The following lemma give bounds for a trajectory entirely contained in the
inner component relative to a monogonal trajectory. Notice that the result
holds as long as the bound on combinatorial lengths of simple trajectories on
$X$ (possibly zero curvature gap) is finite.

######  Lemma 5.7.

We consider a flat sphere $X$ with a monogonal trajectory $\gamma$ of interior
angle $\alpha$ and length $L$. Assuming that the bound $l$ on combinatorial
lengths of simple trajectories is finite, any trajectory $t$ contained in
$X_{\gamma}^{-}$ satisfies one of the following statements:

  1. (1)

$t$ does not contain any monogonal trajectory that would be isotopic to
$\gamma$;

  2. (2)

the combinatorial length of $t$ is at most $2l$. Besides, its metric length is
at most $\frac{L}{\cos(\alpha/2)}$.

###### Proof.

We assume that $t$ contains a monogonal trajectory $\gamma^{\prime}$ that is
isotopic to $\gamma$. Let $\mathcal{C}$ be the topological cylinder cut out by
$\gamma\cup\gamma^{\prime}$. The universal cover $\tilde{\mathcal{C}}$ is
bounded by an inner arc (by developing $\gamma^{\prime}$) and an outer arc (by
developing $\gamma$). Since every interior angle of the inner arc is strictly
bigger than $\pi$, it follows that the maximal extension in both directions of
monogonal trajectory $\gamma^{\prime}$ leaves $\mathcal{C}$ by crossing
$\gamma$ (see Figure 10).

Figure 10. The maximal extension of $\gamma^{\prime}$ crosses $\gamma$.

We draw $\tilde{\mathcal{C}}$ inside the universal cover of
$\mathbb{C}^{\ast}$ and identify the origin with the center of the rotation of
the holonomy of the flat metric along loop $\gamma$ (and also
$\gamma^{\prime}$). It follows that the maximal extension of $\gamma^{\prime}$
is contained in a chord of a disk of radius $\frac{L}{2\cos(\alpha/2)}$ (see
Section 3.2). Consequently, the metric length of $t$ is at most
$\frac{L}{\cos(\alpha/2)}$.

The behaviour of trajectories in a flat annulus (the case of $\mathcal{C}$ is
similar) is described in Section 3.1. The maximal extension of
$\gamma^{\prime}$ in each direction is simple, see Figure 2. It follows that
$t$ is formed by at most two simple trajectories. Since the combinatorial
length of a simple trajectory is at most $l$ by the condition, the desired
bound follows immediately. ∎

Now we are able to generalize the bound of Proposition 5.2 to self-
intersecting trajectories.

######  Proposition 5.8.

We consider a flat sphere $X$ with $n$ conical singularities and a fixed
Delaunay triangulation. Assuming that the bound $l$ on combinatorial lengths
of simple trajectories in $X$ is finite. For a trajectory $t$ in $X$, the
combinatorial length $s$ of $t$ satisfies $s\leq 4l(n-1)\sqrt{|t\cap t|}+4l$
where $|t\cap t|$ is the self-intersection number of trajectory $t$.

###### Proof.

We can decompose trajectory $t$ into finitely many intervals such that at
least $\frac{1}{2l}s-1$ of these trajectories have a combinatorial length
strictly larger than $2l$. We denote by $t_{1},\dots,t_{m}$ these trajectories
contained in $t$ and have $m\geq\frac{1}{2l}s-1$. We are going to give an
upper bound on $m$ in terms of $\sqrt{|t\cap t|}$.

Since combinatorial lengths of $t_{1},\dots,t_{m}$ are larger than $2l$, we
deduce from our assumption that each trajectory $t_{i}$ contains a monogonal
trajectory $\gamma_{i}$ (see Section 3.2). Besides, Lemma 5.7 proves that for
any pair $i,j$ with $i\neq j$, one of two statements holds:

  * •

$t_{i}$ intersects $t_{j}$;

  * •

$\gamma_{i}$ and $\gamma_{j}$ are isotopically distinct.

For each subset $S\subset\\{{1,\dots,m\\}}$ whose number of elements $|S|$ is
at least $2n-2$, there is at least a pair $i,j\in S$ with $i\neq j$ such that
$t_{i}$ and $t_{j}$ intersect. Indeed, if there is no such pair, we deduce
that monogonal trajectories $(\gamma_{i})_{i\in S}$ form a family of $2n-2$
isotopically disjoint loops, in contradiction with Lemma 5.6.

There are $\left(\begin{array}[]{l}m\\\ 2n-2\end{array}\right)$ many choices
of $S$. For any pair $i,j\in S$ with $i\neq j$, there are
$\left(\begin{array}[]{l}m-2\\\ 2n-4\end{array}\right)$ many $S$ containing
this pair. It follows that $|t\cap t|$ is at least:

$\frac{\left(\begin{array}[]{l}m\\\
2n-2\end{array}\right)}{\left(\begin{array}[]{l}m-2\\\
2n-4\end{array}\right)}=\frac{m(m-1)}{(2n-2)(2n-3)}\geq\left(\frac{m-1}{2n-2}\right)^{2}.$

Hence, $m\leq 2(n-1)\sqrt{|t\cap t|}+1$ and $s\leq 2lm+2l\leq
4l(n-1)\sqrt{|t\cap t|}+4l$. ∎

###### Proof of Theorem 1.2.

We are going to prove an upper bound on the number of saddle connections
crossing exactly $s$ Delaunay edges. Since there is at most one saddle
connection in each homotopy class of topological arcs, an oriented saddle
connection $\gamma$ is completely characterized by:

  * •

the triangle corner where $\gamma$ starts;

  * •

for each $1\leq i\leq k-1$, whether the trajectory leaves the triangle through
the left or the right side after the $i$-th crossing with a Delaunay edge.

Forgetting the orientation of the saddle connection, we deduce that there are
at most $(3n-6)2^{s-1}$ saddle connections crossing exactly $s$ Delaunay
edges.

Summing these upper bounds (including the number of saddle connections
coinciding with a Delaunay edge), we obtain that the total number of saddle
connections crossing at most $s$ saddle connections is at most $(3n-6)2^{s}$.

Proposition 5.8 provides the upper bound on the number $s$ of crossings with
Delaunay edges for a saddle connection with at most $k$ self-intersections,
which is $s\leq 4l(n-1)\sqrt{|t\cap t|}+4l$. Proposition 5.2 implies that the
bound $l$ on combinatorial lengths of simple trajectories is at most
$\frac{5n}{\delta}$. Hence, $s\leq\frac{20n(n-1)\sqrt{k}+20n}{\delta}$. When
$k=0$, the bound is given by Corollary 5.4. ∎

###  5.3. Bounding metric lengths of trajectories

We have an upper bound on combinatorial lengths of trajectories with bounded
number of self-intersections (see Proposition 5.8). Combined with a control on
the maximal length of Delaunay edges (see Lemma 4.3), we get an explicit upper
bound on the maximal length of a trajectory with bounded number of self-
intersections (including simple trajectories). Remarkably, this bound does not
depend on the systole (length of the smallest simple saddle connection) but
only on the curvature gap. The bound is uniform on the moduli space of flat
sphere with prescribed conical angles.

###### Proof of Theorem 1.3.

Following Lemma 4.3, the length $L$ of any Delaunay edge in such a flat sphere
$X$ satisfies $L^{2}<\frac{4}{\pi}+\frac{1}{2\pi\delta}$ and thus
$L<\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}}$.

For a simple trajectory $t$ of $X$ that is not contained in a Delaunay edge,
we deduce from the proof of Proposition 5.2 that $t$ is formed by at most
$\frac{5n}{\delta}$ segments, each being contained in a unique Delaunay
triangle. We obtain immediately that the metric length of $t$ is less than
$\frac{10n}{\delta\sqrt{\pi}}+\frac{5n}{\delta^{3/2}\sqrt{2\pi}}$.

For a self-intersecting trajectory $t$ with at most $k$ self-intersections,
Proposition 5.8 implies that the combinatorial length is at most
$4l(n-1)\sqrt{|t\cap t|}+4l$ and Proposition 5.2 implies that
$l\leq\frac{5n}{\delta}$. Combining with
$L<\frac{2}{\sqrt{\pi}}+\frac{1}{\sqrt{2\pi\delta}}$, the metric length is
less than
$\frac{40n(n-1)\sqrt{k}+40n}{\delta\sqrt{\pi}}+\frac{20n(n-1)\sqrt{k}+20n}{\delta^{3/2}\sqrt{2\pi}}.$
∎

## References

  * [1] J. Athreya, Y. Cheung, H. Masur.  Siegel-Veech transforms are in $L^{2}$.  Journal of Modern Dynamics, Volume 14, 1-19, 2019.
  * [2] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Möller.  Strata of $k$-differentials.  Algebraic Geometry, Volume 6, Issue 2, 196–233, 2019.
  * [3] A. Basmajian.  Universal length bounds for non-simple closed geodesics on hyperbolic surfaces.  Journal of Topology, Volume 6, Issue 2, 513–524, 2013.
  * [4] H. Kneser.  Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten.  Jahresbericht der Deutschen Mathematikver-Vereinigung, Volume 38, 248-260, 1930.
  * [5] H. Masur and J. Smillie.  Hausdorff dimension of sets of nonergodic measured foliations.  Annals of Mathematics, Volume 134, Issue 3, 455-543, 1991\.
  * [6] G. Tahar.  Counting saddle connections in flat surfaces with poles of higher order.  Geometriae Dedicata, Volume 196, Issue 1, 145-186, October 2018.
  * [7] W. Thurston.  Shapes of polyhedra and triangulations of the sphere.  Geometry and Topology Monographs, Volume 1, 511-549, 1998.
  * [8] M. Troyanov.  Les surfaces euclidiennes à singularités coniques.  Enseign. Math., Volume 32, 79-94, 1986.


Hopf surfaces in locally conformally Kähler manifolds with potential
Liviu Ornea111Partially supported by CNCS UEFISCDI, project number PN-II-ID-
PCE-2011-3-0118., and Misha Verbitsky222Partially supported by RSCF grant
14-21-00053 within AG Laboratory NRU-HSE.
Keywords: locally conformally Kähler, potential, Hopf manifold, Vaisman
manifold. 2000 Mathematics Subject Classification: 53C55.

Abstract

An LCK manifold with potential is a quotient $M$ of a Kähler manifold $X$
equipped with a positive plurisubharmonic function $f$, such that the
monodromy group acts on $X$ by holomorphic homotheties and maps $f$ to a
function proportional to $f$. It is known that a compact $M$ admits an LCK
potential if and only if it can be holomorphically embedded to a Hopf
manifold. We prove that any non-Vaisman, compact LCK manifold with potential
contains a complex surface (possibly singular) with normalization
biholomorphic to a Hopf surface $H$. Moreover, $H$ can be chosen non-diagonal,
hence, also not admitting a Vaisman structure.

###### Contents

  1. 1 Introduction: LCK manifolds
  2. 2 The form $\omega_{0}$ on a compact LCK manifold with potential
  3. 3 Hopf surfaces in LCK manifolds with potential
    1. 3.1 Complex surfaces of Kähler rank 1
    2. 3.2 Algebraic groups and the Jordan-Chevalley decomposition
    3. 3.3 Finding surfaces in LCK manifolds with potential

##  1 Introduction: LCK manifolds

Let $(M,I)$ be a complex manifold, $\dim_{\mathbb{C}}M\geqslant 2$. It is
called locally conformally Kähler (LCK) if it admits a Hermitian metric $g$
whose fundamental 2-form $\omega(\cdot,\cdot):=g(\cdot,I\cdot)$ satisfies

$d\omega=\theta\wedge\omega,\quad d\theta=0,$ (1.1)

for a certain closed 1-form $\theta$ called the Lee form.

Definition (1.1) is equivalent to the existence of a covering $\tilde{M}$
endowed with a Kähler metric $\Omega$ which is acted on by the deck group
$\operatorname{Aut}_{M}(\tilde{M})$ by homotheties. Let

$\chi:\operatorname{Aut}_{M}(\tilde{M})\longrightarrow{\mathbb{R}}^{>0},\quad\chi(\tau)=\frac{\tau^{*}\Omega}{\Omega},$
(1.2)

be the group homomorphism which associates to a homothety its scale factor.

For definitions and examples, see [DO] and our more recent papers.

An LCK manifold $(M,\omega,\theta)$ is called Vaisman if $\nabla\theta=0$,
where $\nabla$ is the Levi-Civita connection of $g$. The main example of
Vaisman manifold is the diagonal Hopf manifold ([OV3]). The Vaisman compact
complex surfaces are classified in [Be].

Note that there exist compact LCK manifolds which do not admit Vaisman
metrics. Such are the LCK Inoue surfaces, [Be], the Oeljeklaus-Toma manifolds,
[Kas], and the non-diagonal Hopf manifolds, [OV3].

It is known that on any Vaisman manifold, the following formula holds, [Va2],
[DO]:

$d^{c}\theta=\theta\wedge\theta^{c}-\omega.$ (1.3)

Moreover, one can see, [Ve], that the (1,1)-form $\omega_{0}:=-d^{c}\theta$ is
semi-positive definite, having all eigenvalues111The eigenvalues of a
Hermitian form $\eta$ are the eigenvalues of the symmetric operator $L_{\eta}$
defined by the equation $\eta(x,Iy)=g(Lx,y)$. positive, but one which is 0.

An LCK manifold is called with potential if it admits a Kähler covering on
which the Kähler metric has a global and positive potential function which is
acted on by holomorphic homotheties by the deck group. Among the examples: all
Vaisman manifolds, but also non-Vaisman ones, such as the non-diagonal Hopf
manifolds, [OV1], [OV4]. On the other hand, there exist compact LCK manifolds
which cannot admit LCK potential, e.g. Inoue surfaces (see [OV1]) and their
higher dimensional analogues, the Oeljeklaus-Toma manifolds, see 3.3.

One can prove, [OV4], that on a compact manifold, a positive, automorphic
potential can always be deformed to a proper positive, automorphic potential.
The existence of such a potential is equivalent with the image of the
character $\chi$ being isomorphic with ${\mathbb{Z}}$. In this case, the LCK
manifold with potential is called of LCK rank 1.

As on the Kähler covering of an LCK manifold with potential, one has
$\pi^{*}\omega=\psi^{-1}dd^{c}\psi$, where the potential is $\psi=e^{-\nu}$
and the Lee form is $\pi^{*}\theta=d\nu$, one can prove ([OV2], also [AD]):

Claim 1.1:  Equation (1.3) is satisfied on LCK manifolds with potential.

The aim of this paper is to prove that any compact non-Vaisman LCK manifold
with potential contains a complex surface (possibly singular) with
normalization biholomorphic to a Hopf surface which is non-diagonal (this is
the same as being non-Vaisman). As an application, we show that a compact LCK
manifold with potential $(M,\omega,\theta)$ is Vaisman if and only if the form
$d^{c}\theta$ is sign semi-definite.

##  2 The form $\omega_{0}$ on a compact LCK manifold with potential

In general, on an LCK manifold $M$ with potential $\psi$ on $\tilde{M}$, the
norm (w.r.t. the LCK metric) of the Lee form $d\psi$ is not constant. The
constancy of the norm of the Lee form is equivalent to the LCK metric being
Gauduchon (see 2 below) and Vaisman, as shown in [MM].

Definition 2.1: On a complex manifold of complex dimension $n$, a Hermitian
metric whose Hermitian 2-form $\omega$ satisfies the equation
$\partial\overline{\partial}\omega^{n-1}=0$ is called Gauduchon.

Remark 2.2: On a compact Hermitian manifold, a Gauduchon metric exists in each
conformal class and it is unique up to homothety. Moreover, it is
characterized by the co-closedness of its Lee form. A Vaisman metric is a
Gauduchon metric in its conformal class, [G].

Proposition 2.3:  Let $(M,\omega,\theta)$ be a compact LCK manifold with
potential. Then the LCK form $\omega$ is Gauduchon if and only if
$|\theta|=\operatorname{\text{\sf const}}$.

Proof: The Hermitian form $\omega$ is Gauduchon if and only if
$dd^{c}\omega^{n-1}=0$.

We compute $dd^{c}\omega^{n-1}$ using equation (1.3) which is satisfied on an
LCK manifold with potential (1). This gives

$dd^{c}\omega^{n-1}=(n-1)^{2}\omega^{n-1}\wedge\theta\wedge\theta^{c}+(n-1)\omega^{n-1}\wedge
d\theta^{c}.$

On the other hand,

$\omega^{n-1}\wedge\theta\wedge\theta^{c}=\frac{1}{n}|\theta|^{2}\omega^{n}$

and

$d\theta^{c}\wedge\omega^{n-1}=-\omega\wedge\omega^{n-1}+\theta\wedge\theta^{c}\wedge\omega^{n-1}=\left(\frac{1}{n}|\theta|^{2}-1\right)\omega^{n}.$

All in all we get:

$dd^{c}\omega^{n-1}=\frac{(n-1)^{2}}{n}|\theta|^{2}\omega^{n}+(n-1)\left(\frac{1}{n}|\theta|^{2}-1\right)\omega^{n}=(n-1)\big{(}|\theta|^{2}-1\big{)}\omega^{n}.$

Then $dd^{c}\omega^{n-1}=0$ if and only if $|\theta|=1$. This finishes the
proof.

Observe now that the eigenvalues of $\omega_{0}=-d^{c}\theta$ are $1$ (with
multiplicity $n-1$) and $1-|\theta|^{2}$. As $\omega_{0}$ is exact on a
compact manifold, its top power cannot be sign-definite (Stokes theorem). Two
possibilities occur:

  1. 1.

$|\theta|$ is non-constant, and then $1-|\theta|^{2}$ has to change sign on
$M$;

  2. 2.

$|\theta|=\operatorname{\text{\sf const}}.$ and then $|\theta|=1$ and
$\omega_{0}$ is semi-positive definite.

We obtained the following corollary.

Corollary 2.4:  Let $(M,\omega,\theta)$ be a compact LCK manifold with
potential. Then the LCK metric is Gauduchon if and only if
$\omega_{0}=-d^{c}\theta$ is semi-positive definite.

Remark 2.5: Our interest in studying the form $\omega_{0}$ on compact LCK
manifolds arose from the attempt to clarify the relation between the
pluricanonical condition ($(\nabla\theta)^{1,1}=0$, equivalent with
$d^{c}\theta=\theta\wedge\theta^{c}-|\theta|^{2}\omega$), introduced in [Kok],
and the existence of a positive, automorphic potential. In fact, from the
above it can be seen that a compact LCK manifold with potential, and with
constant norm of $\theta$ is pluricanonical, and hence, by [MM], it is
Vaisman.

##  3 Hopf surfaces in LCK manifolds with potential

###  3.1 Complex surfaces of Kähler rank 1

Definition 3.1: ([HL]) A compact complex surface is of Kähler rank 1 if and
only if it is not Kähler but it admits a closed semipositive (1,1)-form whose
zero locus is contained in a curve.

Lemma 3.2: A compact LCK surface $M$ with potential and semi-positive form
$\omega_{0}$ has Kähler rank 1.

Proof: We have to show that $M$ cannot admit a Kähler metric. By absurd, if
$M$ admitted a Kähler form $\Omega$, then, as $\omega_{0}$ is exact,
$\omega_{0}\wedge\Omega=-d(\theta^{c}\wedge\Omega)$ was an exact volume form,
which is impossible by Stokes’ theorem. Hence $M$ is non-Kähler.

Recall that a Hopf surface is a finite quotient of $H$, where $H$ is a
quotient of ${\mathbb{C}}^{2}\backslash 0$ by a polynomial contraction. A Hopf
surface is diagonal if this polynomial contraction is expressed by a diagonal
matrix.

Compact surfaces of Kähler rank 1 have been classified in [CT] and [Br]. They
can be:

  1. 1.

Non-Kähler elliptic fibrations,

  2. 2.

Diagonal Hopf surfaces and their blow-ups,

  3. 3.

Inoue surfaces and their blow-ups.

The LCK Inoue surfaces cannot have LCK metrics with potential, as shown in
[Ot, Corollary 3.13].

A cover of a blow-up of any complex manifold cannot admit plurisubharmonic
functions because, by the lifting criterion, the projective spaces contained
in the blow-up lift to the cover. Thus blow-ups cannot have global potential.

We are left with non-Kähler elliptic fibrations and diagonal Hopf surfaces
which are known to admit Vaisman metrics, see e.g. [Be]. And hence:

Proposition 3.3:  All compact LCK surfaces with potential and with semi-
positive form $\omega_{0}$ admit Vaisman metrics.

For further use it is convenient to list all criteria used to distinguish
Vaisman Hopf surfaces from non-Vaisman ones.

Theorem 3.4:  Let $M$ be a Hopf surface. Then the following are equivalent.

(i)

$M$ is Vaisman.

(ii)

$M$ is diagonalizable.

(iii)

$M$ has Kähler rank 1.

(iv)

$M$ contains at least two distinct elliptic curves.

Proof: Equivalence of the first three conditions is proven above. The
equivalence of (iv) and (ii) is shown by Iku Nakamura and Masahide Kato ([N,
Theorem 5.2]). Note that the cited result refers to primary Hopf surfaces, but
we can always pass to a finite covering and the number of elliptic curves will
not change because the eigenvectors for rationally independent eigenvalues
cannot be mutually exchanged, and if they were dependent, they would produce
infinitely many elliptic curves.

###  3.2 Algebraic groups and the Jordan-Chevalley decomposition

In this section we fix an $n$-dimensional complex vector space $V$.

Lemma 3.5: Let $A\in GL(V)$ be a linear operator, and $\langle A\rangle$ the
group generated by $A$. Denote by $G$ the Zariski closure of $\langle
A\rangle$ in $GL(V)$. Then, for any $v\in V$, the Zariski closure $Z_{v}$ of
the orbit $\langle A\rangle\cdot v$ is equal to the usual closure of $G\cdot
v$.

Proof: Clearly, $Z_{v}$ is $G$-invariant. Indeed, its normalizer $N(Z_{v})$ in
$GL(V)$ is an algebraic group containing $\langle A\rangle$, hence $N(Z_{v})$
contains $G$. The converse is also true: since $\langle A\rangle$ normalizes
$\langle A\rangle\cdot v$, its Zariski closure $G$ normalizes the Zariski
closure $Z_{v}$ of the orbit. Therefore, the orbit $G\cdot v$ is contained in
$Z_{v}$. Since $G\cdot v$ is a constructible set, its Zariski closure
coincides with its usual closure, [H], [Kol]. This gives $\overline{G\cdot
v}\subset Z_{v}$. As $\overline{G\cdot v}$ is algebraic and contains $\langle
A\rangle\cdot v$, the inclusion $Z_{v}\subset\overline{G\cdot v}$ is also
true.

Let $G\subset GL(V)$ be an algebraic group over ${\mathbb{C}}$. Recall that an
element $g\in G$ is called semisimple if it is diagonalizable, and unipotent
if $g=e^{n}$, where $n$ is a nilpotent element of its Lie algebra.

Theorem 3.6: (Jordan-Chevalley decomposition, [H, Section 15]) For any
algebraic group $G\subset GL(V)$, any $g\in G$ can be represented as a product
of two commuting elements $g=g_{s}g_{u}$, where $g_{s}$ is semisimple, and
$g_{u}$ unipotent. Moreover, this decomposition is unique and functorial under
homomorphisms of algebraic groups.

Corollary 3.7:  Let $M$ be a submanifold of a linear Hopf manifold
$H=(V\backslash 0)/A$, $\tilde{M}\subset V\backslash 0$ its
${\mathbb{Z}}$-covering, and $G$ the Zariski closure of $\langle A\rangle$ in
$GL(V)$. Then $\tilde{M}$ contains the $G$-orbit of each point
$v\in\tilde{M}$. Moreover, $G$ is a product of $G_{s}:=({\mathbb{C}}^{*})^{k}$
and a unipotent group $G_{u}$ commuting with $G_{s}$, and bothe of these
groups preserve $\tilde{M}$.

Proof: Let $X$ be the closure of $\tilde{M}$ in ${\mathbb{C}}^{N}$. The ideal
$I_{X}$ of $X$ is generated by polynomials, as shown in [OV1, Proof of Theorem
3.3]. As the polynomial ring is Noetherian, $I_{X}$ is finitely generated,
[AM]. Therefore, $X$ is a cone of a projective variety.

Consider the smallest algebraic group $G$ containing $A$. Then $G$ acts
naturally on $X$ and preserves it. The last assertion of 3.2 is implied by the
Jordan-Chevalley decomposition.

###  3.3 Finding surfaces in LCK manifolds with potential

Lemma 3.8:  Let $M$ be a submanifold of a linear Hopf manifold $H=(V\backslash
0)/A$, $\dim_{\mathbb{C}}M\geqslant 3$, and $G=G_{s}G_{u}$ the Zariski closure
of $\langle A\rangle$ with its Jordan-Chevalley decomposition. Then $M$
contains a surface $M_{0}$, possibly singular, with $G_{u}$ acting non-
trivially on its ${\mathbb{Z}}$-covering $\tilde{M}_{0}\subset V$.

Proof: Another form of this statement is proven by Masahide Kato ([Kat]).

We shall use induction on dimension of $M$. To prove 3.3 it would suffice to
find a subvariety $M_{1}\subset M$ of codimension 1 such that $G_{u}$ acts
non-trivially on its ${\mathbb{Z}}$-covering
$\tilde{M}_{1}\subset{\mathbb{C}}^{n}\backslash 0$. Replacing $V$ by the
smallest $A$-invariant subspace containing $\tilde{M}$, we may assume that the
intersection $\tilde{M}\cap V_{1}\neq V_{1}$ for each proper subspace
$V_{1}\subset V$. Now take a codimension 1 subspace $V_{1}\subsetneq V$ which
is $A$-invariant and such that $G_{u}$ acts on $V_{1}$ non-trivially
(equivalently, such that $A$ acts on $V_{1}$ non-diagonally). Using the Jordan
decomposition of $A$, such $V_{1}$ is easy to construct. Then
$\tilde{M}_{1}^{\prime}:=V_{1}\cap\tilde{M}$ gives a subvariety of $M$ of
codimension 1 and with non-trivial action of $G_{u}$.

The same argument gives the following corollary, also parallel to a theorem by
Ma. Kato.

Corollary 3.9:  Let $M$ be a compact LCK manifold with potential. Then $M$ has
a flag of embedded subvarieties $M\supset M_{1}\supset
M_{2}\supset\cdots\supset M_{\dim M-1}$ with $\operatorname{codim}M_{i}=i$.

Recal that Oeljeklaus-Toma manifolds (see [OT]) do not admit complex curves
([Ver] where the argument doesn’t need smoothness). Then 3.3 implies:

Corollary 3.10:  The Oeljeklaus-Toma manifolds cannot admit LCK structures
with potential.

Lemma 3.11:  Let $M_{1}\subset H=(V\backslash 0)/\langle A\rangle$ be a
surface in a Hopf manifold, possibly singular, and $G=G_{s}G_{u}$ the Zariski
closure of $\langle A\rangle$ with its Jordan-Chevalley decomposition. Assume
that $G_{u}$ acts on the ${\mathbb{Z}}$-covering $\tilde{M}$ non-trivially.
Then the normalization of $M$ is a non-diagonal Hopf surface.

Proof: Replacing $G$ by its quotient by the subgroup acting trivially on
$\tilde{M}$ if necessary, we may assume that $G$ acts properly on a general
orbit in $\tilde{M}$. Then $G$ is at most 2-dimensional. However, it cannot be
1-dimensional because $G_{s}$ contains contractions (hence cannot be
0-dimensional) and $G_{u}$ acts non-trivially. Therefore,
$G_{s}\simeq{\mathbb{C}}^{*}$ and $G_{u}\simeq{\mathbb{C}}$.

Since $G_{s}$ acts by contractions, the quotient $S:=\tilde{M}/G_{s}$ is a
compact curve, equipped with $G_{u}$-action which has a dense orbit. The group
$G_{u}$ can act non-trivially only on a genus 0 curve, and there is a unique
open orbit $O$ of $G_{u}$, with $S\backslash O$ being one point.

Let now $M$ be a normalization of $M_{1}$. Since the singular set of $M_{1}$
is $G_{s}$-invariant, it has dimension at least 1, and since $M$ is normal, it
is non-singular in codimension 1, hence smooth.

All complex subvarieties of $M$ are by construction $G$-invariant, and the
complement of an open orbit is an elliptic curve, hence $M$ has only one
elliptic curve, and is non-diagonalizable by 3.1.

Theorem 3.12:  A non-Vaisman compact LCK manifold $M$ with potential contains
a surface with normalization biholomorphic to a non-diagonal Hopf surface.

Proof: Let $M$ be a compact LCK manifold with potential,
$\dim_{\mathbb{C}}M\geqslant 3$. Then $M$ is holomorphically embedded into a
Hopf manifold ${\mathbb{C}}^{N}\backslash 0/\langle A\rangle$, where
$A\in\mathrm{GL}(N,{\mathbb{C}})$ is a linear operator, see [OV1, Theorem
3.4]. Applying 3.3 and 3.3, we find a non-diagonal Hopf surface in $M$.

As an application, we now prove the following characterization of Vaisman
manifolds:

Corollary 3.13:  Let $(M,I)$ be a compact LCK manifold with potential. Assume
the Hermitian form $\omega_{0}$ is semi-positive definite. Then $(M,I)$ is of
Vaisman type.

Proof: If $\dim_{\mathbb{C}}M=2$, this is just 3.1.

If $\dim_{\mathbb{C}}M\geqslant 3$, $M$ contains a surface whose normalization
is a non-diagonal Hopf surface $H$. Then $\omega_{0}$ restricts to a semi-
positive definite (1,1) form on $H$. By 3.1, $H$ is Vaisman, and hence
diagonal, contradiction.

Acknowledgments. L.O. thanks the Laboratory for Algebraic Geometry at the
Higher School of Economics in Moscow for hospitality and excellent research
environment during February and April 2014, and April 2015.

Both authors are indebted to Paul Gauduchon, Andrei Moroianu, and Victor
Vuletescu for extremely useful disussions.

## References

  * [AD] V. Apostolov, G. Dloussky, Locally conformally symplectic structures on compact non-Kähler complex surfaces, IMRN, 9 (2016), 2717–2747. arXiv:1501.02687.
  * [AM] M.F. Atyiah, I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
  * [Be] F.A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40.
  * [Br] M. Brunella, A characterization of Inoue surfaces Comment. Math. Helv. 88 (2013), no. 4, 859–874. arXiv:1011.2035.
  * [CT] I. Chiose, M. Toma, On compact complex surfaces of Kähler rank one, Amer. J. Math. 135 (2013), no. 3, 851–860. arXiv:1010.2591.
  * [DO] S. Dragomir, L. Ornea, Locally conformally Kähler geometry, Progress in Math. 155, Birkhäuser, 1998.
  * [G] P. Gauduchon, La $1$-forme de torsion d’une variété Hermitienne compacte, Math. Ann. 267 (1984), 495–518.
  * [HL] R. Harvey and H. B. Lawson Jr., An intrinsic characterization of Kähler manifolds, Invent. Math. 74 (1983), no. 2, 169–198.
  * [H] J.E. Humphreys, Linear algebraic groups, GTM 21, 4th ed., Springer, 1998.
  * [Kas] H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds, Bull. London Math. Soc, 45 (2013), no. 1, 15–26. arXiv:1204.1878.
  * [Kat] Ma. Kato, Some Remarks on Subvarieties of Hopf Manifolds, Tokyo J. Math. 2, Nr. 1 (1979), 47–61.
  * [Kok] G. Kokarev, On pseudo-harmonic maps in conformal geometry, Proc. London Math. Soc., 99 (2009), 168–94. arXiv:0705.3821.
  * [Kol] J. Kollár, Exercises in the birational geometry of algebraic varieties. Analytic and algebraic geometry, 495–524, IAS/Park City Math. Ser., 17, Amer. Math. Soc., Providence, RI, 2010.
  * [MM] A. Moroianu, S. Moroianu, On pluricanonical locally conformally Kähler manifolds, IMRN, December 2015 DOI: 10.1093/imrn/rnw151. arXiv:1512.04318.
  * [N] I. Nakamura, On Surfaces of Class VII0 with Curves, Invent. Math. 78, 393–443 (1984).
  * [OT] K. Oeljeklaus, M. Toma, _Non-Kähler compact complex manifolds associated to number fields_ , Ann. Inst. Fourier 55 (2005), 1291–1300.
  * [OV1] L. Ornea, M. Verbitsky, Locally conformal Kähler manifolds with potential. Math. Ann. 348 (2010), 25–33. arXiv:math/0407231.
  * [OV2] L. Ornea, M. Verbitsky, Topology of Locally Conformally Kähler Manifolds with Potential, IMRN, Vol. 2010, pp. 717–726. arXiv:0904.3362.
  * [OV3] L. Ornea, M. Verbitsky, Locally conformally Kahler metrics obtained from pseudoconvex shells, Proc. Amer. Math. Soc. 144 (2016), 325–335, arXiv:1210.2080.
  * [OV4] L. Ornea, M. Verbitsky, LCK rank of locally conformally Kähler manifolds with potential, J. Geom. Physics. 107 (2016), 92–98. arXiv:1601.07413.
  * [Ot] A. Otiman, Morese-Novikov cohomology of locally conformally Kähler surfaces, arXiv:1609.07675.
  * [Va2] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), no. 3, 231–255.
  * [Ve] M. Verbitsky, Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds, Proc. Steklov Inst. Math. 246 no. 3 (2004), 54–78. arXiv:math/0302219.
  * [Ver] S.M. Verbitskaya, Curves on the Oeljeklaus-Toma manifolds, Funct. Anal. Appl. 48, No. 3 (2014), 223–226.

Liviu Ornea
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania, _and_
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Street 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Laboratory of Algebraic Geometry, Faculty of Mathematics,
National Research University Higher School of Economics, 7 Vavilova Str.
Moscow, Russia, _and_
Université Libre de Bruxelles, Département de Mathématique
Campus de la Plaine, C.P. 218/01, Boulevard du Triomphe
B-1050 Brussels, Belgium
<EMAIL_ADDRESS>