Spherical metrics with conical singularities on 2-spheres

TL;DR

This paper investigates the existence of spherical metrics with specified conical singularities on 2-spheres, establishing necessary and sufficient conditions for such metrics when the cone angles are not integer multiples of 2π.

Contribution

It proves the necessity of a known sufficient condition for the existence of spherical metrics with conical singularities, under the assumption that the cone angles are non-integers.

Findings

Established the necessity of the condition for non-integer cone angles.

Extended previous sufficient conditions to necessary ones in specific cases.

Contributed to the classification of spherical metrics with conical singularities.

Abstract

Suppose that $\theta_1,\theta_2,\dots,\theta_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2\pi\theta_1,2\pi\theta_2,\dots,2\pi\theta_n$? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when we assume that $\theta_1,\theta_2,\dots,\theta_n \not\in \mathbb{N}$.

Equations

$$\mathbf{\beta }={\beta }_1{x}_1+\cdots +{\beta }_n{x}_n,$$

$$\chi (S,\mathbf{\beta })=\sum _{i=1}^n{\beta }_i+\chi (S),$$

$$d{s}^2=e^{2u_i}|z_i{|}^{2{\beta }_i}|d{z}_i{|}^2.$$

$$\chi (S,\mathbf{\beta })>0.$$

$$\chi (S,\mathbf{\theta }-1)$$

$$d_1({\mathbb{Z}}_o^n,\mathbf{\theta }-1)$$

$$d_1({\mathbb{Z}}_o^n,\mathbf{\theta }-1)=1,$$

$$({\theta }_i-1)\le \frac{1}{2}\sum _{j=1}^n({\theta }_j-1),\text{for all }0\le i\le n.$$

$$d_S(x^{\prime },x^{\prime \prime })=\text{infimum over the length of piecewise geodesic paths connecting }{x}^{\prime }\text{ and }{x}^{\prime \prime }.$$

$$\underset{x\ne x^{\prime }\in X}{\min }\{d_S(x,x^{\prime })\}\le \pi .$$

$$\chi (S,\mathbf{\theta }-1)$$

$$d_1({\mathbb{Z}}_o^n,\mathbf{\theta }-1)$$

$$U$$

$$l_x=\inf \{l^{\prime }>0\mid x\text{ can be connected to a point in X by a geodesic arc of length l′}\}.$$

$$R=\left\{r\in [0,L)\mid \exists \text{ a local isometry}f:{\Sigma }_r\to S\text{ such that }f(y)=x\right\}$$

$$\dot{F}:\dot{\Sigma }\to S$$

$${\pi }_1(\tilde{S}-\text{singularities})$$

$$\text{length of }\gamma =\text{length of }\tilde{\gamma },$$

Full text

1 Introduction

A classical question in the theory of Riemann surfaces asks which real functions $f$ on a Riemann surface $S$ are equal to the curvature of a pointwise conformal metric. For simplicity, all the surfaces considered in this section are orientable closed 2-manifolds. If $S$ has genus $\geq 2$, then Berger [1] showed that any smooth negative function is the curvature of a unique conformal metric. In the case of a Riemann surface of genus $1$, i.e. when $S$ is a torus with a flat metric $g$, Kazdan and Warner [4] proved that a function $f(\not\equiv 0):S\rightarrow\mathbb{R}$ is the curvature of a metric in the conformal class of $g$ if and only if $f$ changes sign and satisfies $\int_{S}fdA<0$, where $dA$ is the area form $g$.

These results have been generalized by Marc Troyanov in the case of surfaces with singularities of a special type, called conical singularities which we shall define below. Let $S$ be a surface. A real divisor $\boldsymbol{\beta}$ on $S$ is a formal sum

[TABLE]

where $x_{i}\in S$ are pairwise distinct and $\beta_{i}$ are real numbers. For the pair $(S,\boldsymbol{\beta})$, define

[TABLE]

called the Euler characteristic of $(S,\boldsymbol{\beta})$.

Suppose that $\boldsymbol{\beta}=\beta_{1}x_{1}+\dots+\beta_{n}x_{n}$ is a real divisor on $S$ such that $\beta_{i}>-1$. Let $g$ be a Riemannian metric on $S$ defined away from $x_{1},\dots,x_{n}$ such that each point $x_{i}$ has a neighborhood $U_{i}$ in $S$ with coordinate $z_{i}$ satisfying $z_{i}(x_{i})=0$ on which $g$ has the following form,

[TABLE]

Here $u_{i}:U_{i}\rightarrow\mathbb{R}$ is a continuous function such that $u_{i}|_{U_{i}-\{x_{i}\}}$ is differentiable (of class at least $C^{2}$). The point $x_{i}$ is called a conical singularitiy of the metric $g$ of angle $2\pi(\beta_{i}+1)$. We refer to this type of metrics $g$ as (Riemannian) metrics with conical singularities. We say that the metric $g$ (with conical singularities) represents the divisor $\boldsymbol{\beta}$.

In this terminology, Troyanov [8] proved the following two theorems.

Theorem 1** ([8]).**

Let $S$ be a Riemann surface with a real divisor $\boldsymbol{\beta}=\beta_{1}x_{1}+\dots+\beta_{n}x_{n}$ such that $\beta_{i}>-1$, $i=1,\dots,n$. If $\chi(S,\boldsymbol{\beta})<0$, then any smooth negative function on $S$ is the curvature of a unique conformal metric which represents $\boldsymbol{\beta}$.

Theorem 2** ([8]).**

Let $S$ be a Riemann surface with a real divisor $\boldsymbol{\beta}=\beta_{1}x_{1}+\dots+\beta_{n}x_{n}$ such that $\beta_{i}>-1$, $i=1,\dots,n$. If $\chi(S,\boldsymbol{\beta})=0$, then any function $f(\not\equiv 0):S\rightarrow\mathbb{R}$ is the curvature of a conformal metric representing $\boldsymbol{\beta}$ if and only if $f$ changes sign and satisfies $\int_{S}fdA<0$, where $dA$ is the area element of a conformally flat metric on $S$ with singularities.

Theorem 1 generalizes Berger’s result, and Theorem 2 generalizes Kazdan and Warner’s result. Restricting our attention only to constant curvature metrics with conical singularities, we formulate the following question.

Question 1**.**

Let $S$ be a Riemann surface with a real divisor $\boldsymbol{\beta}=\beta_{1}x_{1}+\dots+\beta_{n}x_{n}$ such that $\beta_{i}>-1$, $i=1,\dots,n$. Can $\boldsymbol{\beta}$ be represented by a conformal metric of constant curvature?

The case when $\boldsymbol{\beta}=0$ is completely understood due to the classical uniformization theorems. If $\chi(S)\geq 0$, then any conformal class has a representative of constant curvature. When $\chi(S)<0$, existence and uniqueness of such a conformal metric is provided by classical uniformization theorems of Koebe and Poincaré.

When $\boldsymbol{\beta}\neq 0$, the Theorems 1 and 2 give complete understanding in the case when $\chi(S,\boldsymbol{\beta})\leq 0$. See also the work of McOwen [5] for the $\chi(S,\boldsymbol{\beta})<0$ case.

Here we focus on the particular case when $S$ is the Riemann sphere, which has been least understood. Given a real divisor $\boldsymbol{\beta}=\beta_{1}x_{1}+\dots+\beta_{n}x_{n}$ with $\beta_{i}>-1$, $i=1,\dots,n$, does there exist a conformal metric $g$ with conical singularities on $S$ which represents $\boldsymbol{\beta}$? If we further assume that $g$ has constant curvature $1$, then Gauss-Bonnet Theorem gives a restriction on $\boldsymbol{\beta}$, namely

[TABLE]

But further restrictions on $\boldsymbol{\beta}$ were found in addition to this in the case $n=2,3$. When $n=2$, we simply need to require that $\beta_{1}=\beta_{2}$ by the work of Troyanov [7]. When $n=3$, by Eremenko’s result in [2], a conformal metric with conical singularities exists if and only if some inequalities on the numbers $\beta_{1},\beta_{2},\beta_{3}$ are satisfied. Moreover, in this case, there exists a spherical triangle with angles $\pi(\beta_{1}+1),\pi(\beta_{2}+1),\pi(\beta_{3}+1)$. Here a spherical triangle means a topological closed disk with a metric of constant curvature 1 such that the boundary is a piecewise geodesic loop with three singular points. Examples include geodesic triangles immersed in ordinary sphere $\mathbb{S}^{2}$. A sphere with three conical singularities is the double of such a triangle. When $n>3$, the situation becomes more complicated.

Therefore, before answering the Question 1 for higher $n$’s, perhaps one needs to have a complete list real divisors $\boldsymbol{\beta}$ which may be represented by a spherical metric $g$ on a sphere with conical singularities. Here spherical means that the metric has constant curvature $1$. Note that the divisor $\boldsymbol{\beta}$ can be represented by a spherical metric with conical singularities if and only if there exists a spherical metric with $n$ conical singularities of angles $2\pi(\beta_{1}+1),\dots,2\pi(\beta_{n}+1)$.

Let $\mathbb{R}_{+}^{n}$ be the set of all points in $\mathbb{R}^{n}$ with positive coordinates. A point $\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{n})\in\mathbb{R}_{+}^{n}$ is called admissible if there exists a sphere $S$ with a spherical metric $g$ with $n$ conical points $x_{1},\dots,x_{n}$ of angles $2\pi\theta_{1},\dots,2\pi\theta_{n}$, respectively. In this case, the real divisor $\boldsymbol{\beta}=(\theta_{1}-1)x_{1}+\dots+(\theta_{n}-1)x_{n}$ is represented by $g$. By abuse of notation, we write $\boldsymbol{\theta}-\boldsymbol{1}=\boldsymbol{\beta}$, where $\boldsymbol{1}=(1,\dots,1)$.

Question 2**.**

Which points $\boldsymbol{\theta}\in\mathbb{R}^{n}_{+}$ are admissible?

A major progress was done to answer this question by Mondello and Panov [6], as stated in the next two theorems.

Theorem 3** ([6]).**

If $\boldsymbol{\theta}\in\mathbb{R}^{n}_{+}$ is admissible, then

[TABLE]

where $d_{1}$ is the standard $\ell^{1}$ distance on $\mathbb{R}^{n}$ and $\mathbb{Z}_{o}^{n}$ is the set of all points $\boldsymbol{v}\in\mathbb{R}^{n}$ with integer coordinates such that $d_{1}(\boldsymbol{v},\boldsymbol{0})$ is odd. Moreover, if the equality in (H) is attained, then the holonomy of the a metric which corresponds to $\boldsymbol{\theta}$ is coaxial.

Here we clarify what we mean by saying that the holonomy of a metric with conical sigularities is coaxial. Note that a metric $g$ on $S$ with conical singularities is actually defined as a Riemannian metric only on $S-\{\text{conical sigularities}\}$. Thus we have a holonomy representation of the metric $g$ which is a homomorphism $\phi:\pi_{1}(S-\{\text{conical sigularities}\})\rightarrow SO(3)$ from the fundamental group of $S-\{\text{conical sigularities}\}$ to the group of rotations of the standard sphere $\mathbb{S}^{2}$. The metric $g$ is said to have coaxial holonomy if the image of $\phi$ is contained in an one-parameter subgroup of $SO(3)$.

The following is a partial converse to Theorem 3.

Theorem 4** ([6]).**

If $\boldsymbol{\theta}\in\mathbb{R}^{n}_{+}$ satisfies the positivity constraints (P) and the holonomy constraints (H) strictly, then $\boldsymbol{\theta}$ is admissible. Moreover, each metric corresponding to $\boldsymbol{\theta}$ has non-coaxial holonomy.

Theorems 3 and 4 classified all the admissible points which do not satisfy (H’), where (H’) is the equality in (H),

[TABLE]

but did not provide answer in the complementary case. We formulate this case in the following question.

Question 3**.**

For $n\geq 4$, which points in $\mathbb{R}_{+}^{n}$ satisfying (P) and (1) together are admissible?

In Theorem 5, by restricting to the “non-integral” case, we give an answer to this question. We show that when $\theta_{1},\dots,\theta_{n}\not\in\mathbb{N}$, the sufficient conditions in Theorem 4 are actually necessary.

The “integral” case has been analyzed by Kapovich [3] who showed that a spherical metric with conical singularities of angles $2\pi\theta_{1},\dots,2\pi\theta_{n}$, where $\theta_{1},\dots,\theta_{n}\in\mathbb{N}$, exists if and only if the integers $\theta_{1},\dots,\theta_{n}$ satisfy (1) together with the polygon inequality

[TABLE]

A well-formulated collection of necessary and sufficient conditions on the complement of these two cases (the “mixed” case) is unknown to the author.

Acknowledgement

I thank my advisor, Prof. Michael Kapovich, for introducing this problem to me. I am grateful for his constant encouragement, helpful comments and numerous discussions on this project. I also thank an anonymous referee for commenting on an earlier version of this paper which helped to improve the proof of the main result.

2 Main Result

The aim of this paper is to establish the following theorem.

Theorem 5**.**

Suppose that $\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{n})\in\mathbb{R}_{+}^{n}$ satisfies (P) and (H’) and $n\geq 3$. If $\theta_{i}\not\in\mathbb{N}$ for all $1\leq i\leq n$ then $\boldsymbol{\theta}$ is not admissible.

The case $n=3$ has already been treated in [2].

By a singular spherical surface, we mean a surface, possibly with boundary, with a spherical metric $g$ with a set $X$ of conical singularities. The points in $X$ are called singular points and the points in $S-(\partial S\cup X)$ are called regular points. We further assume that the boundary, if nonempty, is a union of smooth curves. A path $\gamma$ in $S$ is called a geodesic arc if it’s restriction to the complement of $X$ is a connected geodesic arc in the Riemannian sense. We allow geodesic arcs to have singular endpoints. If a geodesic arc has same initial and final point, we call it a geodesic loop. A composition of piecewise geodesic arcs is called a peicewise geodesic path. The metric $g$ induces a distance function $d_{S}$ defined by

[TABLE]

We prove,

Theorem 6**.**

Suppose that $S$ is a singular spherical surface without boundary with a discrete set of conical singularities $X$. Suppose that the metric $d_{S}$ is complete. For $x\in X$, if $S$ has no geodesic loop based at $x$ of length shorter than $\pi$, then

[TABLE]

If the equality occurs, then $S$ is compact and $|X|\leq 2$.

Combining Theorem 5 with Theorems 3 and 4, we get necessary and sufficient conditions on $(\theta_{1},\dots,\theta_{n})$, provided $\theta_{i}\not\in\mathbb{N}$, for which there exists a sphere $S$ with $n\geq 3$ conical singularities of angles $2\pi\theta_{1},\dots,2\pi\theta_{n}$. The analogous case when $n\leq 2$ is known from [7], as discussed in the previous section.

Theorem 7**.**

Suppose that $\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{n})\in\mathbb{R}_{+}^{n}$ where $n\geq 3$. If $\theta_{i}\not\in\mathbb{N}$, $1\leq i\leq n$, then $\boldsymbol{\theta}$ is admissible if and only if

[TABLE]

where $d_{1}$ is the standard $\ell^{1}$ distance on $\mathbb{R}^{n}$ and $\mathbb{Z}_{o}^{n}$ is the set of all points $\boldsymbol{v}\in\mathbb{R}^{n}$ with integer coordinates such that $d_{1}(\boldsymbol{v},\boldsymbol{0})$ is odd.

3 Proof of Theorem 6

Let $\Sigma$ be a closed, genus zero, singular spherical surface with two conical singularities $y$ and $y^{\prime}$ of angle $\theta$. Such a surface has a shape of a football, which we call a “Troyanov’s football”. We denote the closed $r$-neighborhood of $y$ by $\Sigma_{r}$. For small $r$, $\Sigma_{r}$ is a model neighborhood of a conical singularity of angle $\theta$. Note that when $r\geq\pi$, $\Sigma_{r}=\Sigma$.

Definition 8**.**

Let $f:S_{1}\rightarrow S_{2}$ be a map between two singular spherical surfaces. Such a map $f$ is called a locally isometric map (or a local isometry) between these surfaces if the following conditions are satisfied.

$f$ maps the singular (resp. regular) points of $S_{1}$ to the singular (resp. regular) points of $S_{2}$. 2. 2.

$f$ is a local isometry on $S_{1}-(\partial S\cup X)$ in the Riemannian sense. 3. 3.

For each boundary point $s_{1}\in\partial S_{1}$, there exists a neighborhood $U$ of $s_{1}$, a neighborhood $V$ of $s_{2}$, isometries $\phi_{1}:U\rightarrow\mathbb{S}^{2}$, $\phi_{2}:V\rightarrow\mathbb{S}^{2}$ and $\hat{f}\in\text{Isom}(\mathbb{S}^{2})$ such that the following square commutes:

[TABLE]

$f$ is called an isometric embedding if it is an injective locally isometric map.

Let $S$ be a singular spherical surface without boundary with a discrete set $X$ of conical singularities such that the underlying metric space structure of $S$ is complete. Let $x\in X$ be a singular point. We define

[TABLE]

Note that in the definition of $l_{x}$, we also allow geodesic loops based at $x$. If $S$ has has only one singular point $x$ and no geodesic loops based at $x$, i.e. when the definition of $l_{x}$ given above becomes void, then we make the convention $l_{x}=\pi$.

Let $\Sigma$ be a Troyanov’s football with singular points $y$ and $y^{\prime}$ of angles equal to the angle of $S$ at $x$. Our goal is the following: We show that, for any $r<l_{x}$, there exists a local isometry $f:\Sigma_{r}\rightarrow S$ which sends $y$ to $x$. When $l_{x}\geq\pi$, we argue that $S$ can have at most two conical singular points.

Proposition 9**.**

There exists a family of local isometries $f_{r}:\Sigma_{r}\rightarrow S$, $f_{r}(y)=x$, indexed by $r\in[0,\min\{l_{x},\pi\})$, such that, for $s>r$, $f_{s}|_{\Sigma_{r}}=f_{r}$.

Proof.

Let $L=\min\{l_{x},\pi\}$. We start by showing that the set

[TABLE]

is equal to $[0,L)$.

We prove that $R$ is a closed subset of $[0,L)$. Let $(r_{k})_{k\in\mathbb{N}}$ be an increasing sequence in $R$ which converges to some number $r$ in $[0,L)$. Let $f_{r_{k}}:\Sigma_{r_{k}}\rightarrow S$, $f_{r_{k}}(y)=x$, be a sequence of local isometries indexed by $k\in\mathbb{N}$. We can extend the domain of $f_{r_{k}}$ to $\Sigma_{r}$ by keeping $f_{r_{k}}$ constant along the geodesics in $\Sigma_{r}-\text{int }(\Sigma_{r_{k}})$ orthogonal to the boundary $\partial\Sigma_{r_{k}}$. Therefore, we have an equicontinuous family of maps $f_{r_{k}}:\Sigma_{r}\rightarrow S$, for $k\in\mathbb{N}$. Using Arzelá-Ascoli Theorem, by passing to a subsequence we can assume that the sequence $(f_{r_{k}})$ converges to a limit $f:\Sigma_{r}\rightarrow S$. In the following, we prove that $f$ is a local isometry which in turn proves that $R$ is closed.

We first prove that $f$ is a local isometry in the interior of $\Sigma_{r}$. We take any point $p\neq y$ in the interior of $\Sigma_{r}$. Let $q=\lim_{k\rightarrow\infty}f_{r_{k}}(p)\in S$. Since $0<d_{S}(x,q)<L$, $q$ is a regular point of $S$, and hence, there exists $\pi/2>\delta>0$ such that the $\delta$-neighborhood $V$ of $q$ in $S$ is a spherical disk of radius $\delta$. We can choose $k_{0}\in\mathbb{N}$ and an $\delta/2$-neighborhood $U$ of $p$ in $\Sigma$ such that, for $k\geq k_{0}$, $U\subset\Sigma_{r_{k}}$ and $d_{S}(q,f_{r_{k}}(p))<\delta/2$. We may also assume that $U$ is isometric a spherical disk. It is clear that, for $k\geq k_{0}$, $U$ maps into $V$ under $f_{r_{k}}$. Since $V$ has radius $<\pi/2$, $V$ does not contain any closed geodesic. Moreover, $U$ is geodesically convex. Therefore, $f_{r_{k}}|_{U}$ is an isometry onto it’s image. As a result, $f|_{U}$ is an isometric embedding into $S$.

We now show that $f$ is a local isometry at any boundary point $c\in\partial\Sigma_{r}$. The boundary $C_{r}=\partial\Sigma_{r}$ is a round circle in $\Sigma$. Let $c\in C$ and $U$ be an $\delta$-neighborhood of $c$ in $\Sigma$, for some $\delta<\min\{r,L-r\}$. Note that $f(c)$ is a regular point because we assumed that a geodesic from $x_{i}$ to any singular point has length at least $l_{x}>r$. Let $V$ be the $\delta$-neighborhood of $f(c)$ in $S$. We can also assume that $U$ and $V$ are spherical disks with centers $c$ and $f(c)$ respectively. From above, we know that $f$ maps $U_{1}=U\cap\Sigma_{r}-\partial\Sigma_{r}$ locally isometrically into $V$. We show that $f|_{U_{1}}$ is an isometric embedding. If $U_{1}$ is geodesically convex, which happens precisely when $r<\pi/2$, this follows as before. But this argument fails when $r>\pi/2$. In this case, let $f(t_{1})=f(t_{2})$, for some $t_{1},t_{2}$ in $U_{1}$. We can find a third point $t_{3}$ in $U_{1}$ such that, for $i=1,2$, $t_{i}$ and $t_{3}$ can be connected by an (unnormalized) geodesic segment $\gamma_{i}:[0,1]\rightarrow U_{1}$, $\gamma_{i}(0)=t_{3}$, $\gamma_{i}(1)=t_{i}$. The geodesics $f\circ\gamma_{1}$ and $f\circ\gamma_{2}$ in $V$ both have same initial and final points. Since $df$ is injective at all points of $U_{1}$, $\gamma^{\prime}_{1}(0)=\gamma^{\prime}_{2}(0)$ which then implies, after suitable renormalization, that either $\gamma_{1}\subset\gamma_{2}$ or $\gamma_{1}\supset\gamma_{2}$. Therefore, $t_{1}$ and $t_{2}$ must be joined by a geodesic segment in $U_{1}$, say $\gamma_{3}$. But then $f\circ\gamma_{3}$ is a closed geodesic in $S$ which forces $t_{1}=t_{2}$ because $V$ has radius at most $\pi/2$. As a result, $f|_{U_{1}}$ is injective i.e. an embedding. In this case, $f|_{U_{1}}$ can be extended to an embedding of $U$ in $V$ i.e. $f$ is a local isometry at $c$. Thus $R$ is closed in $[0,L)$.

Next we show that $R$ is also an open subset of $[0,L)$. It is clear that if $r\in R$, then the interval $[0,r]$ is also a subset of $R$. If for all $r\in R$ there exists some $s>r$ such that $s\in R$, then $r$ is an interior point of $R$. This shows that $R$ is an open subset of $[0,L)$.

Lemma 10**.**

Let $f:\Sigma_{r}\rightarrow S$ be a local isometry such that $f(y)=x$. Then, there exists some $s>r$ and a local isometry $\tilde{f}:\Sigma_{s}\rightarrow S$ such that $\tilde{f}|_{\Sigma_{r}}=f$.

Proof.

Let $c\in C_{r}$ be any point in the boundary $C_{r}=\partial\Sigma_{r}$. Since $f$ is a locally isometry, there exists a neighborhood $B$ of $c$ in $\Sigma$ and an embedding $f_{1}:B\rightarrow S$ such that $f|_{B\cap\Sigma_{r}}=f_{1}|_{B\cap\Sigma_{r}}$. The maps $f$ and $f_{1}$ patch together to extend $f$ on the larger domain $\Sigma_{r}\cup B$ and such an extension is unique because our maps are analytic.

Let $c_{1},\dots,c_{p}$ be points on $C_{r}$ with the following properties: (i) Each point $c_{j}$ has a neighborhood $B_{j}$ in $\Sigma$ such that $f$ can be extended to a local isometry $\tilde{f}_{j}:B_{j}\cup\Sigma_{r}\rightarrow S$. (ii) $\bigcup_{j=1}^{p}B_{j}$ covers the circle $C_{r}$. (iii) $(B_{j}\cap B_{i})\cup\Sigma_{r}$ is connected. For example, we could choose $B_{j}$ to be a small disk centered at $c_{j}$. Clearly, (ii) implies that $\Sigma_{r}\cup\bigcup_{j=1}^{p}B_{j}$ contains $\Sigma_{s}$, for some $s>r$. Moreover, the uniqueness of the extensions $\tilde{f}_{j}$ and, (iii) ensure that they can be patched together to form an extension $\tilde{f}:\Sigma_{r}\cup\bigcup_{j=1}^{p}B_{j}\rightarrow S$ of $f$. Then $\tilde{f}|_{\Sigma_{s}}$ is an extension of $f$. ∎

The lemma combined with the fact that $R$ is closed in $[0,L)$ shows that any given local isometry $f_{r}:\Sigma_{r}\rightarrow S$, sending $y$ to $x$, can be extended to a local isometry $f_{s}:\Sigma_{r}\rightarrow S$, for any $L>s>r$. Therefore, given a local isometry $f_{r_{0}}:\Sigma_{r_{0}}\rightarrow S$ sending $y$ to $x$, for $0<r_{0}<L$, we have a sequence of local isometries $f_{r}:\Sigma_{r}\rightarrow S$, $f_{r}(y)=x$, indexed by $r\in[0,L)$ such that $f_{s}|_{\Sigma_{r}}=f_{r}$, for $s\geq r$. This completes the proof of the proposition. ∎

Remark 11*.*

The family of local isometries in the proposition may not be replaced by a family of isometries, as shown in the following example:

Example 12**.**

Let $D$ be the closed upper hemisphere of the standard sphere $\mathbb{S}^{2}$. The boundary of $D$ is a geodesic. Let $x_{1},x_{2},x_{3},x_{4}$ be evenly distributed points on the boundary as shown in the figure below.

By identifying the directed edges $a$ with $c$ and $b$ with $d$, we obtain a torus $T$ with a spherical metric having only one conical singularity $x$ of angle $4\pi$, and in this case, $l_{x}=\pi/2$ because the shortest geodesic loop based at $x$ has length $\pi/2$. Note that a local isometry $f_{r}:\Sigma_{r}\rightarrow S$ fails to be an injective map when $r>\pi/4$.

In the following, we assume that $l_{x}\geq\pi$. Let $\dot{\Sigma}$ denote the punctured sphere $\Sigma-\{y\}$. Using Proposition 9, we have a family of local isometries $f_{r}:\Sigma_{r}\rightarrow S$, $f_{r}(y)=x$, indexed by $r\in[0,\min\{l_{x},\pi\})$, such that, for $s>r$, $f_{s}|_{\Sigma_{r}}=f_{r}$. We construct a map

[TABLE]

by setting $\dot{F}(z)=f_{r}(z)$, where $r=d_{\Sigma}(y,z)$. It is clear that $\dot{F}$ is a local isometry, and hence, a Lipschitz continuous function with Lipschitz constant $1$. Since the codomain of $\dot{F}$ is a complete metric space, $\dot{F}$ can be extended to a map $F:\Sigma\rightarrow S$.

Proposition 13**.**

$F:\Sigma\rightarrow S$* constructed above is a surjective map.*

Proof.

Since $F|_{\Sigma-\{y,y^{\prime}\}}$ is a local isometry, image $S^{\prime}$ of $\Sigma-\{y,y^{\prime}\}$ under $F$ is open in $S$. Adjoining the points $x=F(y)$ and $F(y^{\prime})$ compactifies $S^{\prime}$. $F(y)=x$ is an interior point of the image $F(\Sigma)$. We prove that $F(y^{\prime})$ is also an interior point of the image $F(\Sigma)$. Let $D$ be a neighborhood of $F(y^{\prime})$, homeomorphic to $\mathbb{R}^{2}$. $D\cap S^{\prime}$ is a nonempty open set in $D$ which becomes a closed subset when we adjoin $F(y^{\prime})$. So, $D\cap S^{\prime}$ and complement of $(D\cap S^{\prime})\cup\{F(y^{\prime})\}$ are disjoint open subsets whose union is $D-\{F(y^{\prime})\}$. Since $D-\{F(y^{\prime})\}$ is connected, the complement of $(D\cap S^{\prime})\cup\{F(y^{\prime})\}$ must be empty, i.e. $D\cap S^{\prime}\subset S^{\prime}$. This means that $F(\Sigma)$ contains $D$.

Hence, the image $F(\Sigma)$ is open in $S$. Since $S$ is connected, $F(\Sigma)=S$. ∎

We close this section by completing the proof of Theorem 6.

Proof of Theorem 6.

Suppose that $S$ has no geodesic loop based at $x$ of length $\leq\pi$. If $\displaystyle\min_{x\neq x^{\prime}\in X}\{d_{S}(x,x^{\prime})\}$ $\geq\pi$, then $l_{x}\geq\pi$. The image $F(\dot{\Sigma})$ constructed above is a regular open subset of $S$, i.e. it contains no conical singular point of $S$. From Proposition 13, this image misses at most two points of $S$, namely, $F(x)$ and $F(x^{\prime})$. Therefore, $S$ can have at most two singular points. Compactness of $S$ follows from the surjectivity of $F$. ∎

4 Proof of Theorem 5

Let $S$ be a closed, genus zero surface with a spherical metric $g$ with $n$ conical singularities $x_{1},\dots,x_{n}$ of angles $2\pi\theta_{1},\dots,2\pi\theta_{n}$ respectively, where all $\theta_{i}$’s are non-integers and the tuple $(\theta_{1},\dots,\theta_{n})$ satisfies (1). An important feature of the metric $g$ on $S$ is that the holonomy representation is coaxial, which follows from Theorem 3.

Proposition 14**.**

Let $\gamma$ be a geodesic arc in $S$ with singular endpoints. Then the length of $\gamma$ is an integral multiple of $\pi$. In particular, for $i\geq 2$, $d_{S}(x_{1},x_{i})\geq\pi$ and $S$ has no geodesic loop based at $x_{1}$ of length shorter than $\pi$.

Proof.

We shall use Lemma 2.10 of [6] which states the following.

Lemma 15**.**

Let $S^{\prime}$ be a sphere with a spherical metric with conical singularities having coaxial holonomy. Suppose that ${\gamma}^{\prime}$ is a geodesic arc from $x_{1}^{\prime}$ to $x_{2}^{\prime}$, where $x_{1}^{\prime}$ and $x_{2}^{\prime}$ are distinct conical singularities of angles $\not\in 2\pi\mathbb{N}$. Then the length of ${\gamma}^{\prime}$ is an integral multiple of $\pi$.

The lemma shows that the length of $\gamma$ is an integral multiple of $\pi$ if $\gamma$ has distinct endpoints. So, we can assume that $\gamma$ is a geodesic loop based at $x_{1}$. The complement of $\gamma$ in $S$ is comprised of connected components among which there are exactly two components, say $U_{1}$ and $U_{2}$, whose boundary contains the point $x_{1}$. For $i=1,2$, let $y_{i}\in U_{i}$ be a regular point. Let $\rho:\tilde{S}\rightarrow S$ be the two-fold branched covering, branched at the points $y_{1}$ and $y_{2}$, where $\tilde{S}$ is a closed, genus zero surface. By pulling back the metric of $S$, $\tilde{S}$ gets a natural spherical metric with a set of conical singularities $\rho^{-1}(X\cup\{y_{1},y_{2}\})$ where $X=\{x_{1},\dots,x_{n}\}$. Moreover, the holonomy of $\tilde{S}$ is coaxial which follows from holonomy representation

[TABLE]

where $\phi:\pi_{1}({S}-X\cup\{y_{1},y_{2}\})\rightarrow SO(3)$ is the holonomy representation associated to $S$ with conical singularities $y_{1},y_{2},x_{1},x_{2}\dots,x_{n}$. Here we treat the regular points $y_{1}$ and $y_{2}$ of $S$ as singular points of angle $2\pi$. The loop $\gamma$ can be lifted to a geodesic path $\tilde{\gamma}$ with distinct endpoints which are precisely the pre-images of $x_{1}$. We have

[TABLE]

and the later is an integral multiple of $\pi$ which follows from the lemma. ∎

Proof of Theorem 5.

Suppose that, for $n\geq 3$, $\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{n})\in\mathbb{R}_{+}^{n}-\mathbb{Z}^{n}$ satisfies (P) and (1). If $\boldsymbol{\theta}$ is admissible, then there is a sphere with a spherical metric with conical singularities $x_{1},\dots,x_{n}$ such that the angle at $x_{i}$ of $g$ is $2\pi\theta_{i}$. By Proposition 14, $d_{S}(x_{1},x_{i})\geq\pi$, for $i\geq 1$, and there is no geodesic loop based at $x_{1}$ of length shorter than $\pi$. Using Theorem 6, we get $|X|\leq 2$. This is a contradiction. ∎