# Dihedral monodromy of cone spherical metrics

Quentin Gendron  Centro de Investigación en Matemáticas, Guanajuato, Gto., AP
402, CP 36000, México<EMAIL_ADDRESS>and  Guillaume Tahar  Faculty
of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot,
7610001, Israel<EMAIL_ADDRESS>
###### Abstract.

Among metrics of constant positive curvature on a punctured compact Riemann
surface with conical singularities at the punctures, dihedral monodromy means
that the action of the monodromy group
$\mathcal{M}\subset\operatorname{SO}(3)$ globally preserves a pair of
antipodal points. Using recent results about local invariants of quadratic
differentials, we give a complete characterization of the set of conical
angles realized by some cone spherical metric with dihedral monodromy.

###### Key words and phrases:

Quadratic differentials, conical singularities, positive curvature, dihedral
monodromy, co-axial monodromy

###### Contents

  1. 1 Introduction
  2. 2 From differentials to spherical metrics with dihedral monodromy
  3. 3 Differentials with prescribed singularities
  4. 4 Strengthened Gauss-Bonnet inequality
  5. 5 Spherical structures in higher genus
  6. 6 Spherical structures in genus zero

##  1\. Introduction

For a compact Riemann surface $X$ of genus $g$, a finite set of points
$A_{1},\dots,A_{n}\in X$ and a distribution of angles
$2\pi(a_{1},\dots,a_{n})$, a natural generalization of uniformization problem
is about existence of metric of constant positive curvature in
$X\setminus\\{A_{1},\dots,A_{n}\\}$ that extends to singularities
$A_{1},\dots,A_{n}$ as conical points of prescribed angles.

In this problem, Gauss-Bonnet theorem states that the sum of diffuse and
singular curvature is a topological invariant. Total diffuse curvature is
$2\pi(2g-2+n-\sum a_{i})$. Depending on the sign of this quantity, the metric
should be hyperbolic, flat or spherical in the complement of the
singularities.
If the latter quantity is nonpositive, existence of a hyperbolic or a flat
metric has been obtained by several authors, see [12, 13] for recent
references. On the opposite, when this quantity is positive, existence of a
spherical metric is far from being granted.
Some authors have speculated an equivalence between existence of a metric of
constant positive curvature and a notion of stability of bundles (in the
spirit of Yau-Tian-Donaldson conjecture for Fano manifolds), see [9].

We could gain a better understanding of cone spherical metrics by the study of
metrics with constrained monodromy. In this paper, we focus on a class of
metrics introduced in [8]. A cone spherical metric has dihedral monodromy if
its monodromy group (as a subgroup of $\operatorname{SO}(3)$) globally
preserves a pair of antipodal points. This is equivalent to globally preserve
the dual great circle. Rotations of the monodromy group are rotations around
the axis and rotations of angle $\pi$ around any axis whose antipodal points
belong to the preserved great circle (see Section 2.1). A well-studied
subclass of metrics with dihedral monodromy is made of metrics with co-axial
monodromy, see [1].

In [8], the authors obtain the developing map of a spherical structure with
dihedral monodromy by the integration of a quadratic differential. In this
way, they prove existence of cone spherical metrics with dihedral monodromy
for some distributions of angles. The technical result in their paper (Theorem
1.8) amounts to a characterization of configuration of residues at the poles a
quadratic differential may have. They considered only the case of
differentials with simple zeroes and double poles.
Their work can be extended and completed. Indeed, in recent papers is given a
complete characterization of the configurations of local invariants Abelian
and quadratic differentials can realize, see [5, 6]. Using the solution of
this problem (Question 1.7 in [8]), we are able to give an explicit
characterization of the distributions of angles that can be realized by a cone
spherical metric with dihedral monodromy. The key issue, explained in Section
2.6, is the connection between a subclass of spherical metrics (hemispherical
surfaces introduced in Definition 2.5) and a subclass of quadratic
differentials (totally real Jenkins-Strebel differentials introduced in
Definition 2.9).

The main necessary condition for a distribution of angles to be realized by a
spherical metric with dihedral monodromy is the strengthened Gauss-Bonnet
inequality introduced in Proposition 4.2. This inequality is the classical
Gauss-Bonnet inequality where only the angles belonging to $\pi\mathbb{Z}$ are
considered. We briefly explain why this condition appears. A conical
singularity of the spherical metric either corresponds to a double pole of the
quadratic differential whose residue is determined by the conical angle or
with a singularity of order $k\geq-1$ of the differential. Strengthened Gauss-
Bonnet inequality then follows from the classical results about the degree of
a divisor associated to a quadratic differential.

There are additional obstructions leading to exceptional distributions of
angles that cannot be realized by a spherical metric with dihedral monodromy.
They come from general arithmetic obstructions to existence of quadratic
differentials with integer residues obtained in [5, 6]. The list of
obstructions and the complete characterization of distribution of angles that
can be realized are contained in six theorems.
For spherical metrics in genus zero, the classification is given in Theorems
6.3, 6.5 and 6.10 for the strict dihedral case. We remind of [1, Theorem 1]
for the co-axial case in Theorem 6.1.
For spherical metrics in higher genus, the classification is given in Theorem
5.2 for the strict dihedral case and Theorem 5.3 for the co-axial case.

The organization of the paper is the following:

  * •

In Section 2, we introduce the co-axial and dihedral monodromy classes. We
draw the connection between quadratic differentials and cone spherical metrics
with dihedral monodromy.

  * •

In Section 3, we present the results on quadratic and Abelian differentials
with prescribed singularities.

  * •

In Section 4, we state the strenghtened Gauss-Bonnet inequality which is the
main necessary condition for realization of a distribution of angles by a cone
spherical metric with dihedral monodromy.

  * •

In Section 5, we give a characterization of distribution of angles that can be
realized in a spherical surface of genus $g\geq 1$ with co-axial or dihedral
monodromy (Theorems 5.2 and 5.3), comparing the two classes.

  * •

In Section 6, we state Eremenko’s theorem characterizing distribution of
angles for cone spherical metrics with co-axial monodromy in genus zero.
Comparatively, we state and prove the analogous result for metrics with
dihedral monodromy (Theorems 6.3, 6.5 and 6.10).

##  2\. From differentials to spherical metrics with dihedral monodromy

In this section, we recall the basic statements about a special class of
spherical surfaces. Then we show how they are related to quadratic
differentials and give some background on them.

###  2.1. Spherical structures

A spherical structure on a compact surface with finitely many punctures is an
atlas of charts with values in the standard sphere $\mathbb{S}^{2}$ whose
transition maps belong to $\operatorname{SO}(3)$.

A puncture of a spherical surface is a conical singularity of angle $\theta$
if it is locally isometric to the singularity of a a hemispherical sector.

######  Definition 2.1.

A hemispherical sector of angle $\alpha\in]0;2\pi]$ is a singular surface with
boundary, obtained by considering the sector angle $\alpha$ between two
meridians in the standard half-sphere and identifying these two sides. It has
a conical singularity of angle $\alpha$ and a geodesic boundary of length
$\alpha$.

The definition of hemispherical sectors extends by ramified cover to any angle
$\alpha\in\mathbb{R}_{+}^{\ast}$. They provide a geometric model for every
conical singularity.

Given a spherical structure on a surface $S$, there is a representation of the
fundamental group of the complement of the punctures in $S$ into the group of
linear-fractional transformations. Its image is the monodromy of the spherical
structure. Note that it is a subgroup of $\operatorname{SO}(3)$. A first
natural subclass of spherical metrics is given by metrics with reducible or
co-axial monodromy.

######  Definition 2.2.

A cone spherical metric has co-axial monodromy if its monodromy group fixes an
axis. Equivalently, the monodromy is conjugated to a subgroup of
$\operatorname{SO}(3)$ conjugated to $\operatorname{SO}(2)$.

In this paper, we focus on a class of metrics with slightly more general
monodromy, introduced in [8].

######  Definition 2.3.

A cone spherical metric has dihedral monodromy if its monodromy group globally
preserves a pair of antipodal points. Equivalently, the monodromy is
conjugated to a subgroup of $\operatorname{SO}(3)$ conjugated to
$\mathbb{Z}/2\mathbb{Z}\rtimes\operatorname{SO}(2)$.

Among cone spherical metrics with dihedral monodromy, we distinguish co-axial
monodromy (preserving pointwise the two antipodal points of the axis) and
strict dihedral monodromy (dihedral but not co-axial).

###  2.2. Latitude foliation

The dihedral monodromy preserves a great circle in the sphere. We will refer
to it as the equator. We will also refer to the equatorial net as the locus in
the spherical surface that is mapped to the equator in every chart. Since
monodromy acts by isometries, it also preserves the latitude foliation that
decomposes the sphere into circle of constant latitude. Besides, the absolute
latitude, which is the absolute value of the angular distance of a point of
the sphere to the equator, is also well-defined.

We rephrase the previous paragraph. Given a surface $S$ with a cone spherical
metric with dihedral monodromy, the absolute latitude is the map $\phi\colon
S\rightarrow[0,\frac{\pi}{2}]$ which associates to a point the norm of its
latitude. The preimage of $0$ by $\phi$ is the equatorial net while preimages
of $\frac{\pi}{2}$ are the poles. As we will see, if the monodromy is in fact
co-axial, then the sign of the latitude is globally defined.

It should be noted that the only circle of latitude which is a geodesic is the
equator. The others are just loxodromic paths.

Local geometry of conical singularities induces constraints on their position
in the latitude foliation. A conical singularity $A$ of angle $\alpha$ should
satisfy the following conditions:

  * •

if $\alpha\notin\pi\mathbb{Z}$, then $\phi(A)=\frac{\pi}{2}$;

  * •

if $\alpha\notin 2\pi\mathbb{Z}$, then $\phi(A)\in\\{0;\frac{\pi}{2}\\}$;

In other words, any singularity can be a pole of the latitude foliation.
Otherwise, the angle should be an integer multiple of $\pi$ corresponding to
the number of distinct branches of the foliation that approach the
singularity. If this number is odd, then the singularity automatically belongs
to the equatorial net. Indeed, the latitude circle the singularity belongs to
should be preserved by the nontrivial monodromy of a simple loop around this
singularity.

######  Remark 2.4.

For metrics with co-axial monodromy, the latitude defined in each chart is
preserved by the monodromy and it thus globally defined (not just the absolute
latitude). This implies in particular that every singularity whose angle does
not belong to $2\pi\mathbb{Z}$ should be a pole of the latitude foliation.

###  2.3. Hemispherical surfaces

They are a special class of spherical surfaces with cones with dihedral
monodromy.

######  Definition 2.5.

A hemispherical surface is a closed surface with a cone spherical metric
obtained by the gluing of finitely many hemispherical sectors along their
geodesic boundary. The gluing can identify several boundary points and create
conical singularities whose angle is an integer multiple of $\pi$.

This class of surfaces is an example of spherical surface with dihedral
monodromy.

######  Proposition 2.6.

A hemispherical surface has dihedral monodromy.

###### Proof.

For a spherical surface obtained by the gluing of hemispherical sectors, a
latitude foliation is defined in each sector and extends globally through the
boundary. Monodromy preserves the great circle where the boundary of each
sector lies. ∎

A specific property of hemispherical surfaces is that the latitude of every
singularity is either $0$ or $\frac{\pi}{2}$.

Hemispherical surfaces are really simple to describe. They are completely
characterized by the lengths of the boundary segments and the combinatorics of
the gluing pattern. The equatorial net of a hemispherical surface is the union
of the boundaries of every cylinder. We show in Subsection 2.4 that any
spherical surface with dihedral monodromy can be deformed to a hemispherical
surface.

######  Example 2.7.

The most basic hemispherical surface is obtained from a hemispherical sector
of angle $\alpha$. We divide the boundary into two geodesic segments of length
$\frac{\alpha}{2}$ and glue them on each other. We obtain a spherical surface
with two conical singularities of angle $\pi$ and one singularity of angle
$\alpha$.

###  2.4. Loxodromic projection

In spherical geometry (and navigation), a loxodromic path has constant angle
with circles of latitude. In particular, circles of latitude are loxodromic
paths.

For any angle $\theta$, we can define the loxodromic projection flow
$L_{\theta}$. For any point $x$ of the standard sphere with
$\phi(x)\in[0;\frac{\pi}{2}[$, its loxodromic projection $L_{\theta}(x,t)$ is
the intersection of the loxodromic path of constant angle $\theta$ passing
through $x$ with the circle of latitude $(1-t)\phi(x)$ (with $t\in[0;1]$). In
particular, $L_{\theta}(x,0)=x$ while $L_{\theta}(x,1)$ belongs to the
equator. We define $L_{\theta}(x,1)$ as the loxodromic projection of $x$ to
the equator with angle $\theta$.

######  Proposition 2.8.

For any surface $X$ with a cone spherical metric with dihedral monodromy,
there is an isomonodromic deformation $X_{0\leq t\leq 1}$ such that $X=X_{0}$
and $X_{1}$ is a hemispherical surface.

###### Proof.

The latitude foliation decomposes $X$ into "spherical cylinders" that are
continuous families of circles of latitude. For a generic choice of angle
$\theta$, the loxodromic projection flow does not make any pair of
singularities of theses cylinders collide. In the limit, every "spherical
cylinder" which is not bounded by a pole of the latitude foliation degenerates
to an interval exchange map. Conical angles have not been modified but the
latitude of singularities is then either $0$ or $\frac{\pi}{2}$. ∎

###  2.5. Quadratic differentials and half-translation surfaces

On a Riemann surface $X$, a quadratic differential $q$ is a meromorphic
section of $K^{\otimes 2}_{X}$. Outside singularities of odd order, it writes
locally as the square of a meromorphic $1$-form $\pm\sqrt{q}$.
Antiderivatives of $\pm\sqrt{q}$ are the developing maps of a half-translation
structure. This structure is formed by an atlas in the complement of the
singularities of $q$. This atlas is made of charts to $\mathbb{C}$ whose
transition maps are of the form $z\mapsto\pm z+c$.

A singularity of order $a\geq-1$ corresponds to a conical singularity of angle
$(2+a)\pi$ in the associated flat metric. A double pole is a point at infinity
in an infinite cylinder. Besides, at a double pole, if
$q=(\frac{r_{-2}}{z^{2}}+\frac{r_{-1}}{z}+r_{0}+r_{1}z+\dots)dz^{2}$, then the
quadratic residue at the double pole is $r_{-2}$. It should always be nonzero.
In the cylinder neighboring the double pole of quadratic residue $r$, the flat
cylinder is obtained by identifying the two sides of an half-infinite band by
translation of $\pm\sqrt{r}$, see [10] for details.

We define a special class of quadratic differentials.

######  Definition 2.9.

A quadratic differential $q$ on a Riemann surface $X$ is a totally real
Jenkins-Strebel differential if its associated half-translation surface is
formed by the gluing of finitely many semi-infinite cylinders along segments
of their boundary.

Recall that a period of $q$ is the integral of $\pm\sqrt{q}$ along a path
between two singularities of order $\geq-1$. A period is absolute if both
starting and ending points coincide. The set of absolute periods forms a
subgroup of $\mathbb{C}$. Note that the periods of a totally real Jenkins-
Strebel differential are real. It has exactly one double pole for each
cylinder and the other singularities are conical singularities of angle in
$\pi\mathbb{Z}$. They are either simple poles or zeroes of the differential.

###  2.6. Relation between hemispherical surfaces and totally real Jenkins-
Strebel differentials

There is a construction which associates to any hemispherical surface $S$ a
flat surface $S_{\operatorname{flat}}$. It replaces every hemispherical sector
of angle $2a\pi$ by a semi-infinite cylinder of circumference $a$. The lengths
of the segments in the boundary are preserved (up to a factor $2\pi$).

Conversely, given a half-translation surface defined by a totally real
Jenkins-Strebel differential, we replace each semi-infinite cylinder by q
hemispherical sector. This operation is the inverse of the previous one.

The group of absolute periods of the quadratic differential is the image of
the monodromy group of the hemispherical surface into the rotation group along
the preserved axis (up to a factor $2\pi$).

A totally real Jenkins-Strebel differential $q$ is the global square of
$1$-form if and only if every cylinder of the flat surface can be given a sign
such that every boundary segment bounds a positive and a negative cylinder.
The latter condition implies in particular that singularities should be of
even order (with an angle in $2\pi\mathbb{Z}$).
Translated in the language of spherical geometry, a hemispherical surface has
globally defined latitude if and only if its totally real Jenkins-Strebel
differential is the global square of a $1$-form. This property is equivalent
to co-axial monodromy.

###  2.7. Some background on complex projective structures and quadratic
differentials

In this section, we recall some facts about complex projective structures and
their relation with quadratic differentials. This gives interesting background
on this work but will not be used in the remaining sections.

A complex projective structure is defined by charts with values in
$\mathbb{CP}^{1}$ and transition maps in $\operatorname{PSL}(2,\mathbb{C})$.
In particular, a spherical structure is a complex projective structure whose
monodromy is conjugated to a subgroup of $\operatorname{SO}(3)$.

A complex projective structure has dihedral monodromy if it globally preserves
a pair of points of $\mathbb{CP}^{1}$, see Section 6 in [4]. Up to
conjugation, we can assume these preserved points are $\\{{0,\infty\\}}$. Then
the monodromy will act by functions of the form $z\mapsto az^{\pm 1}$ with
$a\in\mathbb{C}^{\ast}$.
This directly implies existence of a quadratic differential $q$ such that any
developing map of the structure is of the form $e^{\int^{z}\sqrt{-q}}$.
Besides, in order to avoid irregular singularities of the developing map, it
is reasonable to restrict to quadratic differentials with at worse double
poles (logarithmic singularities).

In this class of projective structures, the monodromy preserves a metric if
and only if $a\in\operatorname{U}(1)$ for the transition maps. In other words,
periods of the quadratic differential should be real (see Section 1.1 of [8]
for details). In other words, the quadratic differential should be Jenkins-
Strebel.

Then, totally real Jenkins-Strebel quadratic differential introduced in
definition 2.9 are those for which singularities either belong to the equator
or the poles of the latitude foliation. The whole correspondence is summarised
in Table 1.

Complex-analytic object | Geometric structure
---|---
Quadratic differential | Complex projective structure with dihedral monodromy
Jenkins-Strebel quadratic differential | Cone spherical metric with dihedral monodromy
Totally real Jenkins-Strebel differential | Hemispherical surface
Table 1. Correspondence between analytic and geometric objects.

##  3\. Differentials with prescribed singularities

In this section, we recall some results of [5, 6] about existence of
differentials with prescribed local behaviour. Moreover, we recall an
important operation on differentials called the contraction flow.

###  3.1. Quadratic differentials

By abuse of notation, we refer to a pair $(X,q)$ where $X$ is a Riemann
surface of genus $g$ and $q$ a quadratic differential on $X$ as a quadratic
differential. The moduli spaces of these objects are stratified according to
the orders of the singularities of $q$. In this paper, quadratic differentials
have at worse double poles. For $a_{1},\dots,a_{n}\in 2\mathbb{N}^{\ast}$ and
$b_{1},\dots,b_{m}\in 2\mathbb{N}-1$, the stratum of quadratic differentials
with zeroes and simple poles of multiplicities
$a_{1},\dots,a_{n},b_{1},\dots,b_{m}$ and $p$ double poles is denoted
$\mathcal{Q}(a_{1},\dots,a_{n},b_{1},\dots,b_{m},-2^{p})$. See [7] for
details.

We also require that quadratic differentials are not global squares of
$1$-forms. Such differentials are called primitive. Riemann-Roch theorem (or
Gauss-Bonnet theorem) implies that $\sum a_{i}+\sum b_{j}-2p=4g-4$.

For such quadratic differentials, classical complex analysis shows that the
only local invariants up to biholomorphism are the order of the singularity
and the quadratic residue in the case of a double pole, see [10] for details.
At a double pole, every quadratic differential is (up to a biholomorphic
change of variables) $\left(\frac{rdz}{z}\right)^{2}$, where $r^{2}$ is the
quadratic residue.

In Theorems 1.1, 1.2, 1.3 and 1.9 of [6] has been proved a complete
characterization of configurations of quadratic residues that can be realized
in any given stratum of quadratic differentials. We extract from these results
the following two theorems.

######  Theorem 3.1.

Let $\mathcal{Q}$ be a stratum
$\mathcal{Q}(a_{1},\dots,a_{n},b_{1},\dots,b_{m},-2^{p})$ of primitive
quadratic differentials on a surface of genus $g\geq 1$ such that $p\geq 1$.
Every configuration of quadratic residues
$(r_{1},\dots,r_{p})\in(\mathbb{R}_{+}^{\ast})^{p}$ is realized by a quadratic
differential of $\mathcal{Q}$ with the exception of configurations
$(r,\dots,r)$ for strata $\mathcal{Q}(4s,-2^{2s})$ and
$\mathcal{Q}(2s+1,2s-1,-2^{2s})$ in genus one.

In genus zero, it should be noted that a quadratic differential is primitive
if and only if at least one of its singularities is of odd order.

######  Theorem 3.2.

Let $\mathcal{Q}$ be a stratum
$\mathcal{Q}(a_{1},\dots,a_{n},b_{1},\dots,b_{m},-2^{p})$ of primitive
quadratic differentials on the Riemann sphere with $m,p\geq 1$.
Every configuration of quadratic residues
$(b_{1},\dots,b_{p})\in(\mathbb{R}_{+}^{\ast})^{p}$ is realized by a quadratic
differential of $\mathcal{Q}$ with the following exceptions:

  * •

$\mathcal{Q}(p-2,p-2,-2^{p})$ with $p$ odd and configurations of the form
$(A^{2},B^{2},C^{2},\dots,C^{2})$ with $C=A+B$ or $B=A+C$ and $A,B,C>0$;

  * •

$\mathcal{Q}(p-1,p-3,-2^{p})$ with $p$ even and configurations of the form
$(A^{2},A^{2},B^{2},\dots,B^{2})$ with $A,B>0$;

  * •

$\mathcal{Q}(a_{1},\dots,a_{n},b_{1},b_{2},-2^{p})$ and configurations of the
form $(L\cdot f_{1}^{2},\dots,L\cdot f_{p}^{2})$ with $L>0$,
$f_{1},\dots,f_{p}\in\mathbb{N}^{\ast}$, $\sum f_{j}$ is even and $\sum
f_{j}<2p$;

  * •

$\mathcal{Q}(a_{1},\dots,a_{n},b_{1},b_{2},-2^{p})$ and configurations of the
form $(L\cdot f_{1}^{2},\dots,L\cdot f_{p}^{2})$ with $L>0$,
$f_{1},\dots,f_{p}\in\mathbb{N}^{\ast}$, $\sum f_{j}$ is odd and $\sum
f_{j}\leq\max(b_{1},b_{2})$.

###  3.2. Abelian differentials

Some quadratic differentials are global squares of Abelian differentials
($1$-forms). Therefore, square roots of quadratic residues are globally
defined (up to a common factor). Obstructions to realization of configurations
of residues in any stratum states differently.

We consider the stratum $\mathcal{H}(a_{1},\dots,a_{n},-1^{p})$ of pairs
$(X,\omega)$ where $X$ is a compact Riemann of genus $g$ and $\omega$ is a
meromorphic $1$-form with zeroes of orders $a_{1},\dots,a_{n}$ and $p$ simple
poles. Gauss-Bonnet theorem implies that $\sum a_{i}-p=2g-2$.
We consider real configurations of reals
$(\lambda_{1},\dots,\lambda_{x},-\mu_{1},\dots,-\mu_{y})$ with $x+y=p$, whose
sum is zero (to satisfy Residue theorem) and such that
$\lambda_{1},\dots,\lambda_{x},\mu_{1},\dots,\mu_{y}>0$.

In [5], we show in Theorem 1.1 that in genus at least one, the only
obstruction is the Residue theorem.

######  Theorem 3.3.

In any stratum $\mathcal{H}(a_{1},\dots,a_{n},-1^{p})$ of meromorphic
$1$-forms on a surface of genus $g\geq 1$ with $p\geq 1$, every configuration
of residues $(\lambda_{1},\dots,\lambda_{x},-\mu_{1},\dots,-\mu_{y})$ with
$\sum\lambda_{i}=\sum\mu_{j}$ is realized by a differential in the stratum.

In genus zero, there is an additional arithmetic obstruction, described in
Theorem 1.2 of [5] and Theorem 2 of [1].

######  Theorem 3.4.

In any stratum $\mathcal{H}(a_{1},\dots,a_{n},-1^{p})$ of meromorphic
$1$-forms on a surface of genus zero with $p\geq 1$, every configuration of
residues $(\lambda_{1},\dots,\lambda_{x},-\mu_{1},\dots,-\mu_{y})$ with
$\sum\lambda_{i}=\sum\mu_{j}$ is realized by a differential in the stratum
with the following exception.
If the configuration of residues is of the form $(L\cdot f_{1},\dots,L\cdot
f_{x},-L.g_{1},\dots,-L.g_{y})$ with $L>0$ and
$f_{1},\dots,f_{x},g_{1},\dots,g_{y}$ are integers without nontrivial common
factor, then it can be realized in the stratum if and only if $\sum f_{i}=\sum
g_{j}>\max(a_{1},\dots,a_{n})$.

###  3.3. Contraction flow

In Sections 3.1 and 3.2, we considered quadratic differentials defining flat
surfaces of infinite area (because cylinders around double poles are semi-
infinite). There is an action of $\operatorname{GL}^{+}(2,\mathbb{R})$ on
strata of quadratic differentials. This group acts by postcomposition in the
charts and acts naturally on the periods, see [14] for details.

The contraction flow is a one-parameter flow in
$\operatorname{GL}^{+}(2,\mathbb{R})$ that preserves a direction and contracts
exponentially another. For a quadratic differential defining a flat surface of
infinite area, if the contraction flow contracting a generic direction (a
direction where there is no saddle connection), then it converges to a surface
in the stratum where every (absolute and relative) period belongs to the
preserved direction, see Section 5.4 of [11].
Infinite area hypothesis is necessary because otherwise the area of the flat
surface would shrink along the flow.

If we apply the contraction flow to any differential that realizes a real
configurations of residues (like in Sections 3.1 and 3.2), a generic
contraction flow preserving the horizontal direction will converge to a
totally real Jenkins-Strebel differential.

######  Corollary 3.5.

If a distribution of angles is realized by a cone spherical metric with
dihedral monodromy, it is realized by a hemispherical surface with the same
monodromy.

######  Remark 3.6.

The loxodromic projection defined in Section 2.3 in the spherical counterpart
of the contraction flow.

##  4\. Strengthened Gauss-Bonnet inequality

For a distribution of angles and a given genus, we would like to know if the
distribution of angles is realized by a cone spherical metric with dihedral
monodromy on a surface of genus $g$. Conical singularities with integer or
half-integer angles play a special role.

We distinguish between:

  * •

even conical singularities for which the angle is in $2\pi\mathbb{Z}$;

  * •

odd conical singularities for which the angle is in $\pi(2\mathbb{Z}+1)$;

  * •

non-integer conical singularities for which the angle is not in
$\pi\mathbb{Z}$.

For a cone spherical metric with $n$ conical singularities, we have
$n=n_{E}+n_{O}+n_{N}$. These three terms are respectively the numbers of even,
odd and non-integer conical singularities.

######  Definition 4.1.

We consider distributions of angles
$2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
where the three subfamilies are respectively even ($a_{i}\in\mathbb{Z}$), odd
($b_{i}\in\mathbb{Z}+\frac{1}{2}$) and non-integer angles
($c_{i}\notin\frac{1}{2}\mathbb{Z}$). We define:

  * •

the total sum $\sigma=\sum a_{i}+\sum b_{j}+\sum c_{k}$;

  * •

the maximal integral sum $T$ as the sum of $\sum a_{i}$ and the
$2\lfloor\frac{n_{O}}{2}\rfloor$ biggest numbers among the $b_{j}$ (in
particular, $T\in\mathbb{Z}$).

Obviously, we have $\sigma\geq T$. If there is an even number of odd
singularities, then $T=\sum a_{i}+\sum b_{j}$.

Just like a Gauss-Bonnet inequality is necessary for realization of a
distribution of angles by a cone spherical metric, a necessary condition for
realization by a spherical metric with a dihedral monodromy is that a Gauss-
Bonnet inequality should be fulfilled by the integer singularities alone.

######  Proposition 4.2.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
be a distribution of angles. If this distribution of angles is realized by a
cone spherical metric with dihedral monodromy on a surface of genus $g$, then
it should satisfy strengthened Gauss-Bonnet inequalities:

  * •

$T\geq 2g+n-1$ if $n_{O}$ is even and $n_{N}=0$;

  * •

$T\geq 2g+n-2$ otherwise.

###### Proof.

Following Section 2.3, if such a distribution of angles is realized by a cone
spherical metric with dihedral monodromy, then it can be realized by a
hemispherical surface $S$. In this hemispherical surface, singularities with
non-integer angles are automatically at the poles of the latitude foliation.
Among even and odd singularities, some belong to the equatorial net while some
other are at the poles. If the monodromy is dihedral, then every odd
singularity is also at the pole. We consider the totally real Jenkins-Strebel
differential $q$ corresponding to hemispherical surface $S$.
Quadratic differential $q$ belongs to a stratum
$\mathcal{Q}(d_{1},\dots,d_{s},-2^{t})$ with conical singularities of orders
$d_{1},\dots,d_{s}$ and $t$ double poles. We automatically have
$d_{1}+\dots+d_{s}=4g-4+2t$.
Since an even or an odd singularity is either a conical singularity of $q$ or
a double pole, we have $t\geq n_{N}+(n_{E}+n_{O}-s)$. Consequently, $s+t\geq
n$ and $d_{1}+\dots+d_{s}+2s=4g-4+2t+2s\geq 4g+2n-4$.
Since every other singularity is a double pole, $d_{1}+\dots+d_{s}$ is even
and $\sum(d_{i}-2)\leq T$. This implies $T\geq 2g+n-2$.
If $n_{O}$ is even, $n_{N}=0$ and $T=2g+n-2$, then $\sigma=T$ and the
classical Gauss-Bonnet inequality requires $\sigma>2g+n-2$. ∎

##  5\. Spherical structures in higher genus

We apply the results about quadratic and Abelian differentials with prescribed
residues in order to construct hemispherical surfaces realizing the adequate
distribution of angles.

Using equivalence stating in Section 2.4, we construct hemispherical surfaces
corresponding to totally real Jenkins-Strebel differentials. It should be
noted that in genus at least one, in strata of quadratic and Abelian
differentials, there are several degrees of freedom in addition to the
residues. In other words, if there is a differential that realize some
configuration of real residues, then there is another differential that
realizes it and such that the group of periods of the differential is dense in
$\mathbb{R}$. For such a totally real Jenkins-Strebel differential, the
monodromy of the corresponding hemispherical surface is dense in the subgroup
of $\operatorname{SO}(3)$ preserving a pair of antipodal points (pointwise in
the co-axial case and globally in the dihedral case).

###  5.1. Strict dihedral case

Specific obstruction in genus one for quadratic differentials with prescribed
residues (Theorem 3.1) lead to four exceptional families of distributions of
angles that cannot appear in spherical surfaces with strict dihedral
monodromy.

######  Proposition 5.1.

Four exceptional families of distributions of angles cannot be realized by a
cone spherical metric with strict dihedral monodromy on a torus:

  * •

$(4k+2)\pi,c,\dots,c$ with $2k$ non-integer equal angles
$c\notin\pi\mathbb{Z}$;

  * •

$(2k+3)\pi,(2k+1)\pi,c,\dots,c$ with $2k$ non-integer equal angles
$c\notin\pi\mathbb{Z}$;

  * •

$(4k+2)\pi$ for any integer $k$;

  * •

$(2k+3)\pi,(2k+1)\pi$ for any integer $k$.

###### Proof.

According to Proposition 2.8, if such a distribution of angles is realized by
a cone spherical metric with dihedral monodromy, then it can be done by a
hemispherical surface. In this hemispherical surface, singularities with non-
integer equal angles are automatically at the poles of the latitude foliation.
Following Section 2.6, a hemispherical surface corresponds to a totally real
Jenkins-Strebel differential. Zeroes and simple poles of the quadratic
differential can only be the conical singularities with integer angle.
Besides, since the other singularities are double poles, the sum of their
orders should be even. Consequently, these (primitive) quadratic differentials
are in $\mathcal{Q}(4k,-2^{2k})$ or $\mathcal{Q}(2k+1,2k-1,-2^{2k})$. In both
cases, Theorem 3.1 implies that in these strata, a configuration of uniformly
equal residues cannot be realized. If the distribution of angles contains $2k$
equal non-integer angles $c$, the configuration $c,\dots,c$ cannot be
realized. If there is no non-integer angle, then, the $2k$ double poles of the
quadratic differentials correspond to regular points of the spherical metric
and their quadratic residue is $1$. In this case, it cannot be realized
either. ∎

Outside these four exceptional families, the only condition that should be
satisfied is strengthened Gauss-Bonnet inequality stated in Proposition 4.2.

######  Theorem 5.2.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
be a distribution of angles as in Section 4. Outside obstructions in genus $1$
described in Proposition 5.1, there exists a cone spherical metric with strict
dihedral monodromy on a surface of genus $g\geq 1$ with $n$ conical
singularities of prescribed angles if and only if strengthened Gauss-Bonnet
inequality is satisfied.

###### Proof.

We assume the distribution of angles satisfies strengthened Gauss-Bonnet
inequality (Proposition 4.2) and is not concerned by obstructions of
Proposition 5.1. We are going to prove existence of a cone spherical metric
with strict dihedral monodromy in the category of hemispherical surfaces.
Since strengthened Gauss-Bonnet inequality is satisfied, we can find in the
distribution of angles a subset of $s$ even or odd singularities of angles
$\pi(2+d_{i})$ such that $d_{1}+\dots+d_{s}$ is even number whose value is at
least $4g-4+2(n-s)$ (or strictly bigger if $n_{O}$ is even and $n_{N}=0$).
Then, we work with a nonempty stratum $\mathcal{Q}(d_{1},\dots,d_{s},-2^{t})$
of primitive quadratic differentials on Riemann surfaces of genus $g$.
If $g\geq 2$, any configuration of real positive quadratic residues are
realized in the stratum (Theorem 3.1) and they are also realized by totally
real Jenkins-Strebel differential (see Section 3.3). Therefore, we can
construct the analogous hemispherical surface (see Section 2.4).
If $g=1$, unless the maximal integral sum (see Definition 4.1) is realized by
angles $(4k+2)\pi$ or $(2k+3)\pi,(2k+1)\pi$, the same construction can be
carried out (indeed, in Theorem 3.1, the only strata with obstructions are
$\mathcal{Q}(4k,-2^{2k})$ and $\mathcal{Q}(2k+1,2k-1,-2^{2k})$). In the
remaining cases, the only configurations of residues that cannot be realized
are formed by an even number of identical residues. Consequently, the
remaining conical singularities of the spherical structure that correspond to
double poles of the quadratic differentials should have equal angles. This
rules out the possibility of odd singularities among them. There are thus two
possibilities. Either every double pole has a quadratic residue equal to $1$
(because it corresponds to a regular point of the spherical metric) or they
all correspond to non-integer singularities. These cases are covered by
Proposition 5.1. ∎

###  5.2. Co-axial case

In genus at least one, there is no obstruction to existence of Abelian
differentials with prescribed singularities (outside residue theorem). We
consider distributions of angles $\pi(a_{1},\dots,a_{n},c_{1},\dots,c_{p})$
with $a_{1},\dots,a_{n}\in 2\mathbb{N}+2$ and $c_{1},\dots,c_{p}\notin
2\mathbb{N}+2$.

The following result is analogous to Theorem 1 of [1] concerning spherical
metrics with co-axial monodromy on punctured spheres.

######  Theorem 5.3.

Let $2\pi(a_{1},\dots,a_{n},c_{1},\dots,c_{p})$ be a distribution of angles.
There exists a spherical metric with co-axial monodromy on a surface of genus
$g\geq 1$ with $n+p$ conical singularities of prescribed angles if and only
if:

  * •

there is a sequence of signs $\epsilon_{1},\dots,\epsilon_{p}$ such that
$K=\sum_{j=1}^{p}\epsilon_{j}c_{j}\in\mathbb{N}$;

  * •

$\sum a_{i}-2g+2-n-p-K$ is nonnegative and even if $p\geq 1$;

  * •

$\sum a_{i}-2g+2-n$ is positive and even if $p=0$.

###### Proof.

A distribution of angles is realized by a metric with co-axial monodromy if
and only if it is realized by some hemispherical surface (see Proposition
2.8). The totally real Jenkins-Strebel differential corresponding to such a
hemispherical surface is the square of an Abelian differential (see Sections
2.5 and 2.6).
Therefore, there is a subset of $(a_{1},\dots,a_{n})$ corresponding to
singularities that will belong to the equatorial net (and will be zeroes of
the differential). Without loss of generality, we assume they are the $s$
first singularities of the list. Then, the simple poles of the differential
will be the $n-s$ other integer singularities, the $p$ non-integer
singularities and some additional regular points (simple poles with residues
$\pm 1$). Each of them should be given a sign in such a way they sum to zero
(and satisfy residue theorem). Since there is no obstruction in genus at least
one (Theorem 3.3), we only have to check stratum is nonempty.
If a distribution of angles is realized in such a way an integer singularity
of angle $2a\pi$ counts as a simple pole of the Abelian differential (of
residue $\pm a$), then it can also be realized in such a way this singularity
counts as a zero of the form. Indeed, starting from a configuration of
residues summing to zero realized in stratum $\mathcal{H}$, we consider
stratum $\mathcal{H}^{\prime}$ with $a-1$ additional simple poles and an
additional zero of order $a-1$. We replace the simple pole of residue $a$ by
$a$ simple poles of residue $1$ (up to a change of sign). Consequently, we
just have to consider hemispherical surfaces where every integer singularity
counts as a zero of the Abelian differential.
We need to find an Abelian differential in stratum
$\mathcal{H}(a_{1}-1,\dots,a_{n}-1,-1^{t})$ where $t=\sum a_{i}-n-2-2g$.
Besides, simple poles are of residues $\pm 1$ or $\pm c_{i}$. Since the total
residue is zero, there is a signed sum of $(c_{1},\dots,c_{p})$ which is an
integer $K$. If $p=0$, we just need to have an even number of simple poles
(one half having residue $1$ while the other half has residue $-1$). The
condition amounts to $t=\sum a_{i}-n-2-2g$ being even and positive.
If $p\geq 1$, then $t-p$ should be positive and its value should be at least
$K$ with the same parity (if it is bigger, it should be bigger by an even
number in order to keep a total residue equal to zero with residues
compensating each other). Following Theorem 3.3, there is no obstructions to
realize differentials with prescribed residues in genus at least one so the
condition described above are necessary and sufficient. ∎

###  5.3. Comparison

In genus at least one, almost every distribution of angles that are realized
with co-axial monodromy can also be realized with strict dihedral monodromy.
The exceptions are two of the four families of Proposition 5.1.

######  Proposition 5.4.

Let $2\pi(a_{1},\dots,a_{n},c_{1},\dots,c_{p})$ be a distribution of angles.
If it is realized by a spherical metric with co-axial monodromy on a surface
of genus $g$, then it can also be realized by a metric with strict dihedral
monodromy on a surface of genus $g$ with the following the following two
families of exceptions in genus one (parametrized by $k\in\mathbb{N}^{\ast}$):

  * •

$(4k+2)\pi,c,\dots,c$ with $2k$ equal angles $c\notin\pi\mathbb{Z}$;

  * •

$(4k+2)\pi$.

The latter present obstruction for dihedral monodromy (Proposition 5.1) but
clearly satisfy hypothesis of Theorem 5.3.

###### Proof.

Since the distribution is realized by a metric with co-axial monodromy, we
have $\sum a_{i}\geq 2g-2+n+p$. If $g\geq 2$ and $p\geq 1$, this implies
existence of a metric with strict dihedral monodromy (Theorem 5.2).
If $g\geq 2$ but $p=0$, classical Gauss-Bonnet implies $\sum a_{i}>2g-2+n$ and
thus $\sum a_{i}\geq 2g+n-1$. This also implies existence of the adequate
metric.
Then we consider the genus one case. If $p=0$, then we have $\sum a_{i}>n$.
Theorem 5.2 then implies existence of metric with strict dihedral monodromy
except in the four exceptional cases of Proposition 5.1. Among them, the only
family for which there is obstruction is where there is only one conical
singularity the angle of which is of the form $(4k+2)\pi$.
Then, we assume $g=1$ and $p\geq 1$. We have $\sum a_{i}\geq 2g-2+n+p$.
Therefore, strengthened Gauss-Bonnet inequality is satisfied and Theorem 5.2
implies the existence of a metric with strict dihedral monodromy except in the
cases of Proposition 5.1. Since there is at least one even singularity and
$p\geq 1$, we just have to avoid the case where the distribution of angles is
$(4k+2)\pi,c,\dots,c$ with $2k$ non-integer equal angles
$c\notin\pi\mathbb{Z}$. ∎

##  6\. Spherical structures in genus zero

In punctured spheres, monodromy of cone spherical metrics is generated by
rotations around singularities. In particular, if every singularity is even
(its angle belongs to $2\pi\mathbb{Z}$), then the monodromy of the metric is
trivial and the geometric structure is just a cover of the sphere ramified at
these singularities. This case has been already classified, see Section 7 in
[2].

In the following, we assume at least one conical singularity has a non-integer
angle. If there is only one such singularity, then monodromy is cyclic and
thus automatically co-axial. In the next subsection, we will see that in fact,
this implies existence of a second conical singularity with a non-integer
angle.

###  6.1. Co-axial case

In this section, we consider geometric structures whose monodromy group is
contained in the group of rotations around an axis. In particular, it could be
finite rotation group (if angles of non-integer singularities belong to
$\pi\mathbb{Q}$ for example).

Theorem 1 in [1] gave a complete classification of distribution of angles that
are realized with a co-axial monodromy.

######  Theorem 6.1.

Let $2\pi(a_{1},\dots,a_{n},c_{1},\dots,c_{p})$ be a distribution of angles
with $p\geq 1$. There exists a spherical metric with co-axial monodromy on a
punctured sphere with $n+p$ conical singularities of prescribed angles if and
only if:

  * •

there is a sequence of signs $\epsilon_{1},\dots,\epsilon_{p}$ such that
$K=\sum_{j=1}^{p}\epsilon_{j}c_{j}\in\mathbb{N}$;

  * •

$M=\sum a_{i}+2-n-p-K$ is nonnegative and even;

  * •

if $c_{1},\dots,c_{p}$ are commensurable, an additional arithmetic condition
should hold.

Let $v$ be the vector $(c_{1},\dots,c_{p},1,\dots,1)$ with $M+K$ elements
equal to $1$. If $v$ is of the form $L(b_{1},\dots,b_{p+M+K})$ with $L>0$ and
$b_{1},\dots,b_{p+M+K}$ are integers, then the additional condition is:

$2\max(a_{1},\dots,a_{n})\leq\sum b_{i}.$

The condition about signed sums of non-integer angles means in particular that
there should be at least two non-integer singularities.

A proof analogous to the one of Theorem 5.3 could be given to Theorem 6.1 by
using Theorem 3.4.

###  6.2. Strict dihedral case

In the problem of realization of distribution of angles by a metric with
strict dihedral monodromy, the nature of obstructions depends crucially on the
number of odd singularities.

We first give a restriction on the number of odd singularities.

######  Lemma 6.2.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
be a distribution of angles as in Section 4. If it is realized by a cone
spherical metric with strict dihedral monodromy on the punctured sphere, then
$n_{O}\geq 2$.

###### Proof.

If the distribution is realized, then it can be realized by a hemispherical
surface (Proposition 2.8). The latter corresponds to a primitive quadratic
differential (Section 2.5). In genus zero, primitive quadratic differentials
have singularities of odd order (otherwise they are global squares of
$1$-forms). Besides, they have an even number of singularities of odd order
(since the total order is $-4$). Consequently, there are at least two odd
singularities belonging to the equatorial net of the hemispherical surface. ∎

We now treat the cases according to the number of odd singularities. We begin
with the case with at least $4$ odd singularities, then we treat the case with
$3$ odd singularities and finally with $2$.

####  6.2.1. At least four odd singularities

In the case $n_{O}\geq 4$, there is no additional obstruction to strengthened
Gauss-Bonnet inequality.

######  Theorem 6.3.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
be a distribution of angles as in Section 4. If $n_{O}\geq 4$, there exists a
cone spherical metric with strict dihedral monodromy on a punctured sphere
with $n$ conical singularities of prescribed angles if and only if
strengthened Gauss-Bonnet inequality is satisfied. Besides, the metric can be
chosen in such a way its monodromy group is infinite.

###### Proof.

We assume the distribution of angles satisfies strengthened Gauss-Bonnet
inequality (Proposition 4.2). We prove existence of a cone spherical metric
with strict dihedral monodromy in the category of hemispherical surfaces.
Since strengthened Gauss-Bonnet inequality is satisfied, we can find in the
distribution of angles a subset of $s$ even or odd singularities of angles
$\pi(2+d_{i})$ such that $d_{1}+\dots+d_{s}$ is even number whose value is at
least $4g-4+2(n-s)$ (or strictly bigger if $n_{O}$ is even and $n_{N}=0$).
Besides, we assume there are at least four odd singularities among the $s$
chosen singularities. Then, we consider a nonempty stratum
$\mathcal{Q}(d_{1},\dots,d_{s},-2^{t})$ of primitive quadratic differentials
on the Riemann sphere. There are at least four singularities of odd order in
quadratic differentials of these strata. Therefore, any configuration of real
positive quadratic residues are realized in the stratum (Theorem 3.2) and they
are also realized by totally real Jenkins-Strebel differential (see Section
3.3). Therefore, we can construct the analogous hemispherical surface (see
Section 2.4).
Finally, we have to prove that we can realize the hemispherical surface in
such a way the projection of the monodromy group on the rotation group around
the preserved axis has dense image. For a quadratic differential on the
Riemann sphere with at least four odd singularities, the canonical double
cover ramified at the odd singularities is of genus at least one (Riemann-
Hurwitz formula). Therefore, there are other degrees of freedom than quadratic
residues. Up to a small perturbation, we can assume the group of absolute
periods of the Abelian differential in the cover is dense in $\mathbb{R}$. ∎

####  6.2.2. Three odd singularities

If $n_{O}=3$, there are specific obstructions that require to be handled
separately.

######  Proposition 6.4.

Let $k\in\mathbb{N}$, $l\geq k$ and $\alpha,\beta\notin\pi\mathbb{Z}$. If

$\alpha+\beta=(2l+1)\pi\text{ or }\alpha+(2l+1)\pi=\beta\text{ or
}\beta+(2l+1)\pi=\alpha\,,$

then distribution of angles
$((2k+1)\pi,(2k+1)\pi,(2l+1)\pi,\alpha,\dots,\alpha,\beta)$ with $2k-1$ angles
equal to $\alpha$ cannot be realized by a cone spherical metric with strict
dihedral monodromy on a punctured sphere.

###### Proof.

If such a distribution is realized by a cone spherical metric, then it is also
realized by a hemispherical surface. The latter should have at least two half-
integer singularities on its equatorial net. This implies existence of a
quadratic differential in $\mathcal{Q}(2k-1,2k-1,-2^{2k+1})$ whose quadratic
residues are
$\left(l+\frac{1}{2}\right)^{2},\left(\frac{\alpha}{2\pi}\right)^{2}$ and
$2k-1$ quadratic residues equal to $\left(\frac{\beta}{2\pi}\right)^{2}$. This
configuration is forbidden in Theorem 3.2. ∎

In addition to strengthened Gauss-Bonnet inequality stated in Proposition 4.2
and obstruction of Proposition 6.4, an arithmetic condition should be
satisfied if every conical singularity has a rational angle.

######  Theorem 6.5.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},b_{2},b_{3},c_{1},\dots,c_{n_{N}})$ be a
distribution of angles. We assume $b_{1}\geq b_{2}\geq b_{3}$. Outside
obstructions described in Proposition 6.4, there exists a cone spherical
metric with strict dihedral monodromy on a punctured sphere with $n$ conical
singularities of prescribed angles if and only if:

  * •

strengthened Gauss-Bonnet inequality $T=\sum a_{i}+b_{1}+b_{2}\geq n-2$ holds;

  * •

an additional arithmetic condition described below is satisfied when
$c_{1},\dots,c_{n_{N}}\in\pi\mathbb{Q}$.

If vector $(c_{1},\dots,c_{n_{N}},b_{3},1,\dots,1)$ with $T+2-n$ elements
equal to $1$ is of the form $L(r_{1},\dots,r_{T-n_{E}})$ with $L>0$ and
$r_{1},\dots,r_{T-n_{E}}$ integers, then:

  * •

$\sum r_{i}\geq b_{1}+b_{2}$, if $\sum r_{i}$ is even;

  * •

$\sum r_{i}\geq b_{1}$, if $\sum r_{i}$ is odd.

###### Proof.

We assume the distribution of angles satisfies strengthened Gauss-Bonnet
inequality (Proposition 4.2) and is not concerned by obstructions of
Proposition 6.4. We are going to prove existence of a cone spherical metric
with strict dihedral monodromy in the category of hemispherical surfaces.
Since strengthened Gauss-Bonnet inequality is satisfied, we can find in the
distribution of angles a subset of $s$ even or odd singularities of angles
$\pi(2+d_{i})$ such that $d_{1}+\dots+d_{s}$ is an even number whose value is
at least $4g-4+2(n-s)$. Among them there should be two of the three odd
singularities. Then, we consider a nonempty stratum
$\mathcal{Q}(d_{1},\dots,d_{s},-2^{t})$ of primitive quadratic differentials
on the Riemann sphere.
Then, we have to check if the configuration of residues determined by the
remaining singularities and the additional double poles is realized in the
stratum. If one the four obstructions of Theorem 3.2 holds and one even
singularity of angle $2a_{i}\pi$ is not among the $s$ chosen singularities,
then if we can just take stratum
$\mathcal{Q}(2a_{i}-2,d_{1},\dots,d_{n},-2^{t+a_{i}-1})$. The first two
obstructions do not hold when there is zero of even order in differentials of
the stratum. Concerning the two other obstructions, we add an integer residue
to a configuration in which there is a residue which is the square of an
element of $\mathbb{N}+\frac{1}{2}$. Therefore, the sum of integer weights
increases by at least two. If the initial configuration is not realized in its
stratum, it cannot be worse with the new one. Consequently, we assume every
even singularity is among the $s$ chosen singularities.
Similarly, we prove that we could have chosen $b_{1}$ and $b_{2}$ for the two
odd singularities that will correspond to conical singularities of the flat
metric induced by the quadratic differential. For the two last obstructions,
replacing an odd singularity by one with a bigger order, we add at least one
double pole with quadratic residue equal to $1$, increasing the sum of integer
weights by at least two. The two first obstructions appear only if $n_{E}=0$.
The second one involves a configuration of quadratic residues of the form
$(A^{2},A^{2},B^{2},\dots,B^{2})$ with $A,B>0$ and an even number of $B^{2}$.
Here, the third odd singularity corresponds to a double pole with a quadratic
residue equal to $b_{j}^{2}$. There is no double pole with the same quadratic
residue otherwise there would be a fourth odd singularity in the angle
distribution. Therefore, the second obstruction is not relevant when
$n_{O}=3$. The first obstruction of Theorem 3.2 is about configurations of
quadratic residues of the form $(A^{2},B^{2},C^{2},\dots,C^{2})$ with $C=A+B$
or $B=A+C$ and $A,B,C>0$ in strata $\mathcal{Q}(p-2,p-2,-2^{p})$ with $p$ odd.
If among them, one quadratic residue is equal to $1$ while another is equal to
$b_{j}^{2}$, then the third value is the square of an element of
$\mathbb{N}+\frac{1}{2}$. This would imply a existence of a fourth odd
singularity. Therefore, the obstruction only impacts the case where there is
no additional double pole. Consequently, we cannot reach a configuration
forbidden by this obstruction by replacing an odd singularity by one with a
bigger order.
From what precedes, it appears that if a distribution of angles is realized by
a quadratic differential, then the singularities chosen to be in the
equatorial net of the hemispherical surface (or equivalently the conical
singularities of the flat metric) are the biggest possible. They realize the
maximal integer sum.
Thus, we have to consider stratum
$\mathcal{Q}(2a_{1}-2,\dots,2a_{n_{E}}-2,2b_{1}-2,2b_{2}-2,-2^{t})$. We have
$2t-4=2\sum a_{i}-2n_{E}+2b_{1}+2b_{2}-4$. Consequently, $t=b_{1}+b_{2}+\sum
a_{i}-n_{E}$. Among these double poles, one corresponds to $b_{3}$ and there
are $n_{N}$ non-integer singularities. The others are
$K=t-1-n_{N}=b_{1}+b_{2}+\sum a_{i}-n_{E}-n_{N}-1=T+2-n$. The distribution of
angles is realized by a hemispherical surface if and only if the configuration
of quadratic residues formed by squares of numbers
$c_{1},\dots,c_{n_{N}},b_{3},1,\dots,1$ with $K$ elements equal to $1$ is
realized in the stratum.
The first obstruction of Theorem 3.2 is already settled by eliminating
distributions of angles described in Proposition 6.4. Since $b_{3}$ is not
equal to any other number of the list, the second obstruction of Theorem 3.2
is not relevant. The third and the fourth obstruction are already encompassed
in the arithmetic condition.
Finally, it should be noted that the obtained hemispherical surface has strict
dihedral monodromy. Indeed, there are odd singularities on the equatorial net
as well as on the poles of the latitude foliation. Therefore, the monodromy is
not co-axial. ∎

######  Remark 6.6.

In distribution of angles that are forbidden by the additional arithmetic
condition, there should always be two non-integer singularities with equal
angles (at least two elements among $r_{1},\dots,r_{n_{N}+K+1}$ should be
equal to one). They have smaller angle than any other singularity.

######  Example 6.7.

Distribution of angles $3\pi,3\pi,3\pi,\frac{3\pi}{2},\frac{3\pi}{2}$ cannot
be realized by a metric with dihedral monodromy because of the additional
arithmetic obstruction.

####  6.2.3. Two odd singularities

If $n_{O}=2$, then it has already been proved in Section 4 of [3] that in
absence of non-integer singularities, monodromy is co-axial. Therefore we will
assume $n_{N}\geq 1$.

As previously, some specific obstructions have to be handled separately. They
correspond to the first two obstructions of Theorem 3.2.

######  Proposition 6.8.

For $k\in\mathbb{N}$, $\alpha,\beta\notin\pi\mathbb{Z}$, the following
distributions of angles are not realized by a cone spherical metric with
strict dihedral monodromy on a punctured sphere:

  * •

$((2k+3)\pi,(2k+1)\pi,\alpha,\dots,\alpha,\beta,\beta)$ with $2k$ angles equal
to $\alpha$;

  * •

$((2k+3)\pi,(2k+1)\pi,\alpha,\dots,\alpha)$ with $2k$ angles equal to
$\alpha$;

  * •

$((2k+3)\pi,(2k+1)\pi,\alpha,\alpha)$.

###### Proof.

If such a distribution is realized by a cone spherical metric, then it is also
realized by a hemispherical surface. The latter should have at least two half-
integer singularities on its equatorial net. This implies existence of a
quadratic differential in $\mathcal{Q}(2k+1,2k-1,-2^{2k+1})$ where zeroes are
the two odd singularities of the spherical metric while the double poles
correspond either to non-integer singularities or regular points. Therefore,
quadratic residues can be equal to $\left(\frac{\alpha}{2\pi}\right)^{2}$ or
$\left(\frac{\beta}{2\pi}\right)^{2}$ (if the double pole corresponds to a
non-integer singularity) or are equal to one (if the double pole corresponds
to a regular point of the spherical metric).
In any case, the second obstruction of Theorem 3.2 forbids existence of a
quadratic differential with such a configuration of quadratic residues. ∎

######  Proposition 6.9.

For $k\in\mathbb{N}$, $\alpha,\beta,\gamma\notin\pi\mathbb{Z}$, the following
distributions of angles are not realized by a cone spherical metric with
strict dihedral monodromy on a punctured sphere:

  * •

$((2k+3)\pi,(2k+3)\pi,\alpha,\dots,\alpha,\beta,\gamma)$ with $2k+1$ angles
equal to $\alpha$ and $\alpha=\beta+\gamma$;

  * •

$((2k+3)\pi,(2k+3)\pi,\alpha,\dots,\alpha,\beta,\alpha+\beta)$ with $2k+1$
angles equal to $\alpha$;

  * •

$((2k+3)\pi,(2k+3)\pi,\alpha,\dots,\alpha,\beta)$ with $2k+1$ angles equal to
$\alpha$, $\alpha+\beta=2\pi$, $\beta=\alpha+2\pi$ or $\alpha=\beta+2\pi$;

  * •

$((2k+3)\pi,(2k+3)\pi,\alpha,\alpha+2\pi)$;

  * •

$((2k+3)\pi,(2k+3)\pi,\alpha,\beta)$ with $\alpha+\beta=2\pi$.

###### Proof.

We proceed in the same way as in the proof of Proposition 6.8 except that in
this case we refer to the first obstruction of Theorem 3.2. ∎

In addition to strengthened Gauss-Bonnet inequality (see Proposition 4.2) and
obstructions of Proposition 6.8 and 6.9, an arithmetic condition should be
satisfied if non-integer singularities have commensurable angles.

######  Theorem 6.10.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},b_{2},c_{1},\dots,c_{n_{N}})$ be a
distribution of angles with $n_{N}\geq 1$. We assume $b_{1}\geq b_{2}$.
Outside obstructions described in Proposition 6.8 and 6.9, there exists a cone
spherical metric with strict dihedral monodromy on a punctured sphere with $n$
conical singularities of prescribed angles if and only if the two following
conditions hold:

  * •

strengthened Gauss-Bonnet inequality $T=\sum a_{i}+b_{1}+b_{2}\geq n-2$ holds;

  * •

an additional arithmetic condition described below is satisfied when
$c_{1},\dots,c_{n_{N}}$ are commensurable.

Considering vector $v=(c_{1},\dots,c_{n_{N}},1,\dots,1)$ with $T+2-n$ elements
equal to $1$, if $v$ is of the form $L(r_{1},\dots,r_{T-n_{E}})$ with $L>0$
and $r_{1},\dots,r_{T-n_{E}}$ integers, then:

  * •

$\sum r_{i}\geq b_{1}+b_{2}$, if $\sum r_{i}$ is even;

  * •

$\sum r_{i}\geq b_{1}$, if $\sum r_{i}$ is odd.

###### Proof.

We assume the distribution of angles satisfies strengthened Gauss-Bonnet
inequality (Proposition 4.2) and is not concerned by obstructions of
Proposition 6.8 and 6.9. We are going to prove existence of a cone spherical
metric with strict dihedral monodromy in the category of hemispherical
surfaces.
Since strengthened Gauss-Bonnet inequality is satisfied, we can find in the
distribution of angles a subset of $s$ even or odd singularities of angles
$\pi(2+d_{i})$ such that $d_{1}+\dots+d_{s}$ is an even number whose value is
at least $4g-4+2(n-s)$. Among them there should be the two odd singularities.
Then, we consider a nonempty stratum $\mathcal{Q}(d_{1},\dots,d_{s},-2^{t})$
of primitive quadratic differentials on the Riemann sphere.
Then, we have to check if the configuration of residues determined by the
remaining singularities and the additional double poles is realized in the
stratum. If one the four obstructions of Theorem 3.2 holds and one even
singularity of angle $2a_{i}\pi$ is not among the $s$ chosen singularities,
then if we can just take stratum
$\mathcal{Q}(2a_{i}-2,d_{1},\dots,d_{n},-2^{t+a_{i}-1})$. The first two
obstructions do not hold when there is zero of even order in differentials of
the stratum. Concerning the two other obstructions, we add an integer residue.
In any case, the sum of integer weights increases by at least one while the
bound depends on the orders of the two odd singularities (in other words, the
bound does not change). If the initial configuration is not realized in its
stratum, it cannot be worse with the new one. Consequently, we assume every
even singularity is among the $s$ chosen singularities.

Thus, we have to consider stratum
$\mathcal{Q}(2a_{1}-2,\dots,2a_{n_{E}}-2,2b_{1}-2,2b_{2}-2,-2^{t})$. We have
$2t-4=2\sum a_{i}-2n_{E}+2b_{1}+2b_{2}-4$. Consequently, $t=b_{1}+b_{2}+\sum
a_{i}-n_{E}$. Among these double poles, there are $n_{N}$ non-integer
singularities. The others are $K=t-n_{N}=b_{1}+b_{2}+\sum
a_{i}-n_{E}-n_{N}=T+2-n$. The distribution of angles is realized by a
hemispherical surface if and only if the configuration of quadratic residues
formed by squares of numbers $c_{1},\dots,c_{n_{N}},1,\dots,1$ with $K$
elements equal to $1$ is realized in the stratum.
The first and second obstruction are already settled by eliminating
distributions of angles described in Propositions 6.8 and 6.9. The third and
the fourth obstruction are already encompassed in the arithmetic condition.
Finally, the obtained hemispherical surface cannot have co-axial monodromy
since it has odd singularities on the equator and at least one non-integer
singularity at the poles since $n_{N}\geq 1$. ∎

###  6.3. Comparison

Just like in Proposition 5.4, some distributions of angles that can be
realized by a spherical metric with co-axial monodromy can also be realized by
a spherical metric with strict dihedral monodromy. We give a complete
characterization.

######  Proposition 6.11.

Let $2\pi(a_{1},\dots,a_{n_{E}},b_{1},\dots,b_{n_{O}},c_{1},\dots,c_{n_{N}})$
be a distribution of angles. If it is realized by a spherical metric with co-
axial monodromy on a punctured sphere, then it can also be realized by a
metric with strict dihedral monodromy on a punctured sphere if and only if
$n_{O}\geq 2$ and $n_{O}+n_{N}\geq 3$.

###### Proof.

Lemma 6.2 prove that $n_{O}\geq 2$ is a necessary condition for existence of a
spherical metric with dihedral monodromy on a punctured sphere. Besides, if
the number of singularities with nontrivial monodromy is exactly two, then the
monodromy of the metric is automatically co-axial. Therefore, $n_{O}+n_{N}\geq
3$ is a necessary condition. We will prove that these two conditions are also
necessary.
We consider a distribution of angles realized by a spherical metric with co-
axial monodromy. We assume it satisfies the two necessary conditions. Theorem
6.1 implies in particular that $\sum a_{i}\geq n_{E}+n_{O}+n_{N}-2$. Since we
have $n_{O}+n_{N}\geq 3$, we deduce $n_{E}\geq 1$. Therefore, the distribution
of angles is not forbidden by Propositions 6.4, 6.8 and 6.9. Besides, the
condition on the sum of orders of even singularities in Theorem 6.1 clearly
implies the strengthened Gauss-Bonnet inequality (see Proposition 4.2). It
remains to prove that the distribution of angles satisfies the hypothesis of
Theorems 6.3, 6.5 and 6.10.
If $n_{O}\geq 4$, then there is no additional condition to check and Theorem
6.3 proves that the distribution of angles is realized by a spherical metric
with strict dihedral monodromy.

If $n_{O}=2$ or $n_{O}=3$, we assume $b_{1}\geq b_{2}\geq b_{3}$. We already
know that $\sum a_{i}\geq n-2$. We set $K=T+2-n$ and $T=\sum
a_{i}+b_{1}+b_{2}\geq n-2$ (Strengthened Gauss-Bonnet inequality). Thus,
$K\geq b_{1}+b_{2}$. Theorem 6.5 and 6.10 require an additional arithmetic
condition.
We first consider the case $n_{O}=3$. The condition of Theorem 6.5 is the
following. Let $v$ be the vector $(c_{1},\dots,c_{n_{N}},b_{3},1,\dots,1)$
with $K$ elements equal to $1$. If the distribution of angles cannot be
realized by a metric with strict dihedral monodromy, then $v$ is of the form
$L(r_{1},\dots,r_{n_{N}+K+1})$ with $L>0$ and $r_{1},\dots,r_{n_{N}+K+1}$ are
integers. We should have:

  * •

$\sum r_{i}\geq b_{1}+b_{2}$ if $\sum r_{i}$ is even;

  * •

$\sum r_{i}\geq b_{1}$ if $\sum r_{i}$ is odd.

If the condition is not satisfied, then $L=1$ because otherwise the $K\geq
b_{1}+b_{2}$ elements of $v$ that are equal to $1$ would be enough to satisfy
the bound. However, $b_{3}$ is not an integer, which leads to a contradiction
since $\frac{b_{3}}{L}$ is required to be an integer.
In the case $n_{O}=2$, the condition of Theorem 6.10 is essentially the same.
We have to consider vector $(c_{1},\dots,c_{n_{N}},b_{3},1,\dots,1)$ with $K$
elements equal to $1$. Similarly, if the condition is not satisfied, then
$L=1$. This implies $c_{1},\dots,c_{n_{N}}$ are integers, which is a
contradiction. We already know by hypothesis that $n_{N}\geq 1$. This ends the
proof. ∎

##### Acknowledgements.

The second author is supported by the Israel Science Foundation (grant No.
1167/17) and the European Research Council (ERC) under the European Union
Horizon 2020 research and innovation programme (grant agreement No. 802107).
The second author would also like to thank Boris Shapiro for introducing him
to the field of spherical metrics.

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