# Riemannian Surfaces with Simple Singularities

Marc Troyanov

###### Abstract

In this note we discuss the geometry of Riemannian surfaces having a discrete
set of singular points. We assume the conformal structure extends through the
singularities and the curvature is integrable. Such points are called _simple
singularities_. We first describe them locally and then globally using the
notion of (real) divisor. We formulate a Gauss-Bonnet formula and relate it to
some asymptotic isoperimetric ratio. We prove a classifications theorem for
flat metrics with simple singularities on a compact surface and discuss the
Berger–Nirenberg Problem on surfaces with a divisor. We finally discuss the
relation with spherical polyhedra.

This article is a translation of the paper [15], to be included in the
forthcoming book _Reshetnyak’s Theory of Subharmonic Metrics_ edited by
François Fillastre and Dmitriy Slytskiy and to be published by Springer and
the Centre de recherches mathématiques (CRM) in Montréal.

AMS Mathematics Subject Classification: 53.c20, 52.a55

Keywords: Gauss-Bonnet formula; Riemann-Hurwitz Formula; Uniformization;
Singular Riemannian surface; spherical polyhedra; conical singularities;
Berger-Nirenberg Problem.

##  1 Local Description

Among the singular Riemannian metrics on surfaces, the simplest ones are those
with isolated singularities. Away from a discrete set, these metrics are then
smooth (say of class $C^{2}$). We will make two additional hypotheses which
are natural from a geometric point of view. The first concerns the conformal
structure. If $g$ is a Riemannian metric on a surface $S$ having an isolated
singularity at $p$, and if $U$ is a neighborhood of $p$ homeomorphic to the
disk, then $U^{\prime}=U\setminus\\{0\\}$ has a well-defined conformal
structure (since the metric $g$ is smooth on $U^{\prime}$, one may apply Korn-
Lichtenstein’s Theorem). From the classification of conformal structures on
the annulus, we know that $U^{\prime}$ is conformally equivalent to a standard
annulus

$A_{\rho}=\\{z\in\mathbb{C}\mid\rho<|z|<1\\},$

for some fixed parameter $\rho\in[0,1)$. In this paper, we will always assume
$U^{\prime}$ to be conformally equivalent to the punctured disk $A_{0}$. In
other words, we are assuming that the conformal structure of $U^{\prime}$
extends to $U$ (i.e. the point $p$ is a removable singularity from the
conformal viewpoint).

In particular, every point of $S$ (singular or not) has a coordinate
neighborhood in which the metric $g$ can be written as

$g=\rho(x,y)(dx^{2}+dy^{2})=\rho(z)|dz|^{2},$

where $z=x+iy$ and $\rho$ is a positive function that is of class $C^{2}$
outside the singularities. Such coordinates are called isothermal coordinates.

As a counterexample, one may consider a non-negative function $\phi$ on
$\mathbb{C}$ that vanishes exactly on a contractible compact subset
$Q\subset\mathbb{C}$. Let us denote by $S=\mathbb{C}/Q$ the space obtained by
identifying all the points of $Q$ and $g=\phi(z)|dz|^{2}$. If $Q$ contains
more than one point, then $(S,g)$ has a singular point $q=[Q]$ that does not
satisfy the above condition.

Our second assumption concerns the curvature. It says that if $K$ denotes the
curvature and $dA$ the area element of $g$, then

$\int_{U^{\prime}}|K|dA<\infty.$

An important class of singularities satisfying the above conditions is given
by the simple singularities:

Definition. A conformal metric $g$ on a Riemann surface $S$ is said to have a
simple singularity of order $\beta$ at $p\in S$ if it can be locally written
as

$g=e^{2u(z)}|z|^{2\beta}|dz|^{2},$

where $\beta$ is a real number and $u$ is a function satisfying

$u\in L^{1}\quad\text{and}\quad\Delta u\in L^{1}.$

In this definition, $z=x+iy$ is a local coordinate on $S$ defined in a
neighborhood $U$ of $p$ and such that $z(p)=0$. The Lebesgue space $L^{1}$ is
defined with respect to the Lebesgue measure $dxdy$ on $U$ and the Laplacian
of $u$ is defined in the sense of distributions by $\Delta
u=-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}$.

Simple singularities naturally appear in several contexts as we shall soon
see. A first class of examples is given by the following result due to MacOwen
[9, Appendix B]:

######  Theorem 1.1.

Let $g=e^{2u}|dz|^{2}$ be a conformal metric on the unit disk
$D=\\{z\in\mathbb{C}~\mid|z|<1\\}$ having a singularity at the origin. Suppose
that $g$ is smooth on the punctured disk $D^{\prime}=D\setminus\\{0\\}$. If
there exist $\ell\in\mathbb{R}$ and $a,b>0$ such that the curvature $K$ of $g$
satisfies

$-b|z|^{\ell}\leq K(z)\leq-a|z|^{\ell},$

then $0$ is a simple singularity of $g$.

A simple singularity of order $\beta<-1$ is always at infinite distance while
a simple singularity of order $\beta>-1$ is always at a finite distance of
ordinary points. For a singularity of order $-$1, both cases can occur, see
§2.2 in [5]. A cusp is a simple singularity of order $\beta=-1$ admitting a
neighborhood of finite area.

A simple singularity of order $\beta>-1$ is also called a conical singularity
of (total) angle $\theta=2\pi(\beta+1)$. Such a singularity can indeed be
approximated by a Euclidean cone of total angle $\theta$. In particular, if
the curvature of $g$ is bounded in some neighborhood of a conical singularity,
there exists an “exponential map” making it possible to parametrize a
neighborhood of the conical singularity by a neighborhood of the vertex of its
tangent cone. In other words, one can introduce polar coordinates near a
conical singularity. Moreover, if the curvature is continuous, then these
polar coordinates are of class $C^{1}$ with respect to the isothermal
coordinates, see [16] for proofs of these facts.

##  2 Global description

To investigate singular surfaces having several simple singularities, it is
convenient to introduce the following notion:

Definition. A divisor on a Riemann surface $S$ is a formal sum
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$. The support of this divisor
is the set $\mathrm{supp}(\boldsymbol{\beta})=\\{p_{1},\dots,p_{n}\\}$. A
conformal metric $g$ on $S$ represents the divisor $\boldsymbol{\beta}$ if it
is smooth on the complement of $\mathrm{supp}(\boldsymbol{\beta})$ and if $g$
has a simple singularity of order $\beta_{i}$ at $p_{i}$ for $i=1,\dots,n$.

Examples. (1) The metric $g=|dz|^{2}$ on the Riemann sphere
$\mathbb{C}\cup\\{\infty\\}$ represents the divisor
$\boldsymbol{\beta}=(-2)\cdot\infty$.

(2) More generaly, the metric $g=|z|^{2\alpha}|dz|^{2}$ on
$\mathbb{C}\cup\\{\infty\\}$ represents the divisor
$\boldsymbol{\beta}=\alpha\cdot 0+(-2-\alpha)\cdot\infty$.

(3) If $\omega=\varphi(z)dz$ is a meromorphic differential on the Riemann
surface $S$, then $g=|\omega|^{2}$ is a flat Riemannian metric with simple
singularities representing the divisor
$\boldsymbol{\beta}=\mathrm{div}(\omega)$.

(4) If $(S_{1},g_{1})$ is a smooth Riemannian surface and $f:S\to S_{1}$ is a
branched covering, then $g=f^{*}(g_{1})$ is a Riemannian metric on $S$
representing the ramification divisor of $f$, that is
$\boldsymbol{\beta}=\sum_{p}O_{p}(f)\cdot p$, where $O_{p}(f)$ is the
ramification order of $f$ at $p$ (i.e. the local degree minus $1$).

(5) If $S$ is a two-dimensional polyhedron (euclidean, spherical or
hyperbolic) with vertices $p_{1},\dots,p_{n}$, then the metric induced by the
geometric realization of that polyhedron represents the divisor
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$, where
$2\pi(\beta_{i}+1)=\theta_{i}$ is the sum of the angles at $p_{i}$ of all
faces incident with $p_{i}$.

(6) Let $(\tilde{S},\tilde{g})$ be a smooth Riemannian surface on which a
finite group $\Gamma$ acts by isometries. If $S=\tilde{S}/\Gamma$ is a surface
without boundary, then it inherits a Riemannian metric with simple
singularities representing the divisor
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$, where
$\beta_{i}=(\frac{1}{n_{i}}-1)$. Here, the point $p_{i}$ is the image of a
point $\tilde{p}_{i}\in\tilde{S}$ such that $\Gamma$ has a stabiliser of order
$n_{i}$ at $\tilde{p}_{i}$. This examples generalizes to two-dimensional
orbifolds.

In examples 4 to 6, all the singular points are conical singularities. An
important source of examples, where no singular point is conical, is given by
the following Theorem:

######  Theorem 2.1.

Let $(S^{\prime},g^{\prime})$ be a complete Riemannian surface of class
$C^{2}$ with finite total curvature: $\int_{S^{\prime}}|K|dA<\infty$. Then
there exists a compact Riemann surface $S$, a divisor
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ on $S$ such that
$\beta_{i}\leq-1$ for all $i$, and a conformal metric $g$ on $S$ representing
this divisor such that $(S^{\prime},g^{\prime})$ is isometric to
$(S\setminus\mathrm{supp}(\boldsymbol{\beta}),g)$.

This result is essentially due to A. Huber, we refer to §1.1 and §2.9 in [5]
for a discussion and a proof of Huber’s theorem in the above formulation. It
is not difficult to see that if the surface $(S^{\prime},g^{\prime})$ has
finite area, then $\beta_{i}=-1$ for all $i$.

##  3 Some global geometry

For compact Riemannian surfaces with simple singularities, there is a well
known Gauss-Bonnet Formula (see e.g. [2] for the case where all orders satisfy
$\beta_{i}\leq-1$). To state the Formula we define the Euler characteristic of
a surface $S$ with divisor as
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ as
$\chi(S,\boldsymbol{\beta})=\chi(S)+\sum_{i}\beta_{i}$.

######  Theorem 3.1 (The Gauss-Bonnet Formula).

Let $(S,g)$ be a compact Riemannian surface whose metric represents a divisor
$\boldsymbol{\beta}$. Then the total curvature of $(S,g)$ is finite and we
have

$\frac{1}{2\pi}\int_{S}KdA=\chi(S,\boldsymbol{\beta}).$

See e.g. [5, Theorem 2.8] for a proof.

For example, if $\omega$ is a meromorphic differential on the closed Riemann
surface of genus $\gamma$, then the Gauss-Bonnet Formula implies that the
degree of $\omega$ (i.e. the number of zeroes minus the number of poles), is
equal to $2\gamma-2$. Indeed, $g=|\omega|^{2}$ is a flat metric representing
the divisor $\mathrm{div}(\omega)$. Another application of the Gauss-Bonnet
formula is the Riemann-Hurwitz formula:

######  Proposition 3.2 (The Riemann-Hurwitz Formula).

Let $f:S\to S_{1}$ be a branched cover of degree $d$ between two closed
surfaces, then

$\chi(S)+\sum_{p\in S}O_{p}(f)=d\chi(S_{1}),$

where $O_{p}(f)$ is the branching order of $f$ at $p$.

###### Proof.

Pick an arbitrary smooth metric $g_{1}$ on $S_{1}$ and set $g=f^{*}(g_{1})$.
Then $g$ is a Riemannian metric with simple singularities on $S$ representing
the ramification divisor of $f$. The above formula follows now from Theorem
3.1, since we obviously have

$\int_{S}KdA=d\cdot\int_{S_{1}}K_{1}dA_{1}.$

∎

Huber’s theorem, together with the Gauss-Bonnet Formula, is a refinement of
the Cohn-Vossen inequality:

######  Proposition 3.3 (The Cohn-Vossen inequality).

Let $(S^{\prime},g^{\prime})$ be a compact Riemannian surface of class $C^{2}$
with finite total curvature:
$\int_{S^{\prime}}|K^{\prime}|dA^{\prime}<\infty$, then we have

$\frac{1}{2\pi}\int_{S^{\prime}}K^{\prime}dA^{\prime}\leq\chi(S^{\prime}).$

Moreover we have equality if $(S^{\prime},g^{\prime})$ has finite area.

###### Proof.

Huber’s Theorem tells us that $(S^{\prime},g^{\prime})$ admits a
compactification $(S,g)$ where $g$ is a metric representing a divisor
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ such that $\beta_{i}\leq-1$
for all $i$. We then have from the Gauss-Bonnet Formula

$\frac{1}{2\pi}\int_{S^{\prime}}K^{\prime}dA^{\prime}=\frac{1}{2\pi}\int_{S}KdA=\chi(S)+\sum_{i=1}^{n}\beta_{i}\leq\chi(S)-n=\chi(S^{\prime}).$

If $S^{\prime}$ has finite area, then we have $\beta_{i}=-1$ for all $i$ and
the above inequality is in fact an equality. ∎

Note that although the Cohn-Vossen inequality is a consequence of Huber’s
Theorem 2.1 and the Gauss-Bonnet Formula, one should not consider it to be a
corollary of these results. The reason is that the proof of Huber’s theorem is
in part based on the Cohn-Vossen inequality.

The difference between $\chi(S^{\prime})$ and $\chi(S,\boldsymbol{\beta})$ in
the Cohn-Vossen inequality is an isoperimetric constant:

######  Theorem 3.4.

Let $(S,g)$ be a compact Riemannian surface whose metric $g$ represents a
divisor $\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ such that
$\beta_{i}\leq-1$ for all $i$. Fix a point $q$ on
$S^{\prime}=S\setminus\\{p_{1},\dots,p_{n}\\}$ and denote by $A(q,r)$ the area
of $B_{q}(r):=\\{x\in S^{\prime}\mid d(q,x)\leq r\\}$ and $L(q,r)$ the length
of $\partial B_{q}(r)$. Then

$\lim_{r\to\infty}\frac{L^{2}(q,r)}{4\pi
A(q,r)}=-\sum_{i=1}^{n}(\beta_{i}+1)=\chi(S^{\prime})-\chi(S,\boldsymbol{\beta}).$

This result is due to K. Shiohama [11], but R. Finn obtained a partial result
in this direction [2, Theorem 10].

##  4 Classifying Flat Metrics

Let us now formulate a classification theorem for flat metrics with simple
singularities on a compact surface.

######  Theorem 4.1.

Let $S$ be a compact Riemann surface with divisor
$\boldsymbol{\beta}=\sum\beta_{i}p_{i}$. Then there exists a conformal flat
metric representing $\boldsymbol{\beta}$ on $S$ if and only if
$\chi(S,\boldsymbol{\beta})=0$. Moreover this metric is unique up to
homothety.

This theorem has several proofs, see e.g. [13] and [5, §7]. We give here the
proof from [5].

###### Proof.

Introduce in the neighborhood of each $p_{i}$ a coordinate $z_{i}$ such that
$z_{i}(p_{i})=0$ and choose an arbitrary conformal metric $g_{0}$ on $S$ such
that $g_{0}=|dz_{i}|^{2}$ in the neighborhood of each $p_{i}$. Let us now
choose a positive function $\rho:S\to\mathbb{R}$ which is of class $C^{2}$ on
$S\setminus\mathrm{supp}(\boldsymbol{\beta})$ and such that
$\rho=|z_{i}|^{2\beta_{i}}$ in the neighborhood of each $p_{i}$. The metric
$g_{1}=\rho g_{0}$ is then a conformal metric representing the divisor
$\boldsymbol{\beta}$.

Since the desired metric must be conformal on $S$, it can be written as
$g=e^{2u}g_{1}$. Note that if $u$ is a function of class $C^{2}$ on $S$ such
that

$\Delta_{1}u=-K_{1},$ (1)

where $\Delta_{1}$ and $K_{1}$ denote the Laplacian and the curvature of
$g_{1}$, then $g=e^{2u}g_{1}$ is a flat conformal metric representing
$\boldsymbol{\beta}$ on $S$. Because $\Delta_{1}$ is a singular operator, it
is more convenient to write the previous equation as

$\Delta_{0}u=-\rho K_{1},$ (2)

where $\Delta_{0}$ is the Laplacian of the smooth metric $g_{0}$.

Note that (1) and (2) are equivalent equations, but since $K_{1}$ vanishes in
a neighborhood of the points $p_{i}$ and the functions $\rho$ and $K_{1}$ are
of class $C^{2}$ on $S\setminus\mathrm{supp}(\boldsymbol{\beta})$, the right
hand side of (2) is of class $C^{2}$ on the whole surface $S$.

It is well known that the partial differential equation (2) has a solution if
and only if the integral of the right hand side vanishes, which follows from
the Gauss-Bonnet formula:

$\int_{S}K_{1}\rho
dA_{0}=\int_{S}K_{1}dA_{1}=2\pi\chi(S,\boldsymbol{\beta})=0.$

We have thus proved the existence of a flat conformal metric on $S$
representing the divisor $\boldsymbol{\beta}$. The uniqueness follows from the
fact that if $g_{1}$ and $g_{2}$ are two such metrics, then
$g_{2}=e^{2v}g_{1}$ for a harmonic function $v$ on $S$. This function is
constant (because $S$ is a closed surface) and the two metrics are therefore
homothetic. ∎

The above Theorem gives us a short proof of the Uniformization Theorem for the
sphere:

######  Theorem 4.2.

Any Riemann surface homeomorphic to the two-sphere is conformally equivalent
to $\mathbb{C}\cup\\{\infty\\}$.

###### Proof.

Let us choose a point $p$ in $S$ and consider the divisor
$\boldsymbol{\beta}=(-2)\cdot p$. Observe that
$\chi(S,\boldsymbol{\beta})=2-2=0$. The previous theorem tells us that there
is a conformal flat metric $g$ on $S$ representing this divisor. It is clear
that $(S,g)$ is isometric (and thus conformally equivalent) to
$\left(\mathbb{C}\cup\\{\infty\\},|dz|^{2}\right)$. ∎

Of course the Uniformization Theorem also means that there exists a smooth
conformal metric of constant positive curvature on $S$. However, it is hard to
prove this result by directly solving the corresponding Berger-Nirenberg
problem, that is by directly constructing a conformal metric of curvature
$+1$. The above proof on the other hand is almost trivial.

##  5 The Berger–Nirenberg Problem on Surfaces with Divisors

The classical Berger-Nirenberg problem is the following:

Problem 1. Let $S$ be a Riemann surface and $K:S\to\mathbb{R}$ a function on
this surface. Is there a conformal metric on $S$ whose curvature is the
function $K$? If it exists, is such a metric unique?

This problem is clearly not well posed for open surfaces. One could hope that
the problem is well posed for complete Riemannian metrics, however, it is not
difficult to construct families $\\{g_{\lambda}\\}$ of conformal metrics on a
Riemann surface which are complete, conformal, of the same curvature and whose
geometry at infinity drastically varies with $\lambda$, in the sense that they
are not mutually bilipschitz. An example is given in [4]. The previous
discussion, in particular Huber’s Theorem 2.1, suggests to replace the Berger-
Nirenberg Problem on open surfaces by a version of the problem on compact
surfaces with a divisor.

Problem 2. Let $(S,\beta)$ be a compact Riemann surface with divisor, and
$K:S\to\mathbb{R}$ be a smooth function. Is there a conformal metric $g$ on
$S$ that represents $\boldsymbol{\beta}$ and whose curvature is the function
$K$? If it exists, is such a metric unique?

We have already answered this question when $K$ vanishes everywhere.

Problem 2 is studied in the papers [17] (in the case of conic singularity) and
[5] in the general case. The results can be summarized in a form which is
similar to the classical theory in the smooth case as it is exposed in the
foundational article [6] by Jerry Kazdan and Frank Warner.

######  Theorem 5.1.

Let $(S,\boldsymbol{\beta})$ be a compact Riemann surface with a divisor
$\boldsymbol{\beta}=\sum\beta_{i}p_{i}$ , and $K:S\to\mathbb{R}$ be a smooth
function. Suppose that there exists $p>1$ such that
$h_{i}(z)=|z-p_{i}|^{2\beta_{i}}K(z)$ is a function of class $L^{p}$ in a
neighborhood of each $p_{i}$. Moreover

  1. (a)

If $\chi(S,\boldsymbol{\beta})>0$, we assume $\sup(K)>0$ and
$q\chi(S,\boldsymbol{\beta})<2$, where $q=p/(p-1)$.

  2. (b)

If $\chi(S,\boldsymbol{\beta})=0$, we assume either that $K\equiv 0$ or
$\sup(K)>0$ and $\int_{S}KdA_{0}<0$, where $dA_{0}$ is the area element of a
flat conformal metric representing $\boldsymbol{\beta}$.

  3. (c)

If $\chi(S,\boldsymbol{\beta})<0$, we assume $K\leq 0$ and $K\not\equiv 0$.

Then there exists a conformal metric $g$ on $S$ which represents the divisor
$\boldsymbol{\beta}$ and whose curvature is $K$. In case (c), this metric is
unique.

A very brief idea of the proof is presented in [4] (see [17] and [5] for
details). Some particular cases of this theorem have been obtained previously
by W. M. Ni, R. MacOwen and P. Aviles. At the beginning of the twentieth
century, Emile Picard had already studied the case of curvature $-1$ in [10].
The hypotheses of the previous theorem impose a decay of the curvature when
approaching the singularities of order $<-1$. The next result, only valid for
non-positive curvature, does not impose such a behavior.

######  Theorem 5.2.

Let $S$ be a compact Riemann surface and $g_{1}$ be a conformal metric
representing a divisor $\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ such
that $n\geq 1$ and $\chi(S,\boldsymbol{\beta})<0$. Let $K:S\to\mathbb{R}$ be a
smooth nonpositive function such that

$bK\leq K_{1}\leq aK<0$

on the complement of a compact subset of
$S^{\prime}=S\setminus\\{p_{1},\dots,p_{n}\\}$, where $K_{1}$ is the curvature
of $g_{1}$ and $a,b$ are positive constants. Then there exists a unique
conformal metric $g$ on $S$ which represents $\boldsymbol{\beta}$, has
curvature $K$ and is conformally quasi-isometric to $g_{1}$.

See [5, Theorem 8.1, 8.4] or [9] for the proof. As an application, using this
Theorem, one can construct metrics with prescribed (negative) curvature having
cusps. The previous Theorem also admits a generalization to non-compact
Riemann surfaces of finite type having hyperbolic ends.

We end this section with two results on the Berger-Nirenberg Problem on closed
Riemann surfaces with divisor. The first result, proved in [12] by Junjie
Tang, allows us to solve the Berger-Nirenberg Problem when
$\chi(S,\boldsymbol{\beta})$ is small enough.

######  Theorem 5.3.

Let $S$ be a closed Riemann surface with a divisor
${\boldsymbol{{\boldsymbol{\alpha}}}}=\sum_{i=1}^{n}\alpha_{i}p_{i}$ such that
$\chi(S,{\boldsymbol{\alpha}})=0$, and let us pick a conformal metric $g_{0}$
representing ${\boldsymbol{\alpha}}$.

Suppose another divisor $\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ with
same support is given on $S$, such that $\chi(S,\boldsymbol{\beta})<0$ and
consider a function $K:S\to\mathbb{R}$ such that $K=O(|z-p_{i}|^{\ell_{i}})$
(in the neighborhood of each $p_{i}$), where $\ell_{i}>-2(1+\alpha_{i})$.

_(A)_ If $\beta_{i}\leq\alpha_{i}$ for all $i$, then a necessary condition for
the existence of a conformal metric $g$ on $S$ representing
$\boldsymbol{\beta}$ is

$\int_{S}KdA_{0}<0,$ (3)

where $dA_{0}$ is the area element of $g_{0}$.

_(B)_ There exists $\varepsilon>0$ (depending on $S$, $\alpha$ and $K$) such
that if $\max_{i}|\beta_{i}-\alpha_{i}|\leq\varepsilon$, then (3) is a
sufficient condition for the existence of a conformal metric $g$ representing
$\beta$.

The second result is due to Dominique Hulin [3]. It allows us to solve the
problem when $K$ is close enough to a given non-positive function:

######  Theorem 5.4.

Let $S$ be a compact Riemann surface with a divisor
$\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ such that
$\chi(S,\boldsymbol{\beta})<0$. Let $K_{1},k:S\to\mathbb{R}$ be two smooth
functions on $S$ such that $K_{1}\leq 0\leq k$ everywhere on $S$, and
$K_{1}\not\equiv 0$. Suppose that $|z-p_{i}|^{2\beta_{i}}(k(z)-K_{1}(z))\in
L^{p}$ for some $p>1$ in the neighborhood of the points $p_{i}$. Then there
exists a constant $C>0$ (depending on $S$, $\boldsymbol{\beta}$, $K_{1}$ and
$k$) such that if

$K_{1}\leq K\leq K_{1}+Ck$

on $S$, then there exists a conformal metric $g$ of curvature $K$ on $S$ which
represents $\boldsymbol{\beta}$.

The dependence of the constant $C$ on $S$, $\boldsymbol{\beta}$, $k$ and
$K_{1}$ is explicitly given in [3, Theorem 6.1]

##  6 Spherical Polyhedra

The following result classifies spherical surfaces with less than three
conical points.

######  Theorem 6.1.

Let $g$ be a metric on the sphere $S^{2}$ representing a divisor
$\boldsymbol{\beta}=\beta_{1}p_{1}+\beta_{2}p_{2}$ and whose curvature is
constant $K=+1$. Then $\beta_{1}=\beta_{2}$, moreover:

  1. (a)

If $\beta_{i}$ is not an integer, then $p_{1}$ and $p_{2}$ are conjugate
points (that is $d(p_{1},p_{2})=\pi$).

  2. (b)

If $\beta=m\in\mathbb{N}$, then $(S,g)$ is isometric to a branched covering of
degree $m+1$ of the standard sphere $\mathbb{S}^{2}$, (with its canonical
metric) branched over two points, with ramification order equal to $m$.
Moreover, two such metrics are isometric if and only if their singularities
are of the same order and separated by the same distance.

Let us call _spherical polyhedra_ a Riemannian surface homeomorphic to the
sphere, with conical singularities of order $\beta_{i}\in(-1,0)$ and whose
Gauss curvature is constant $K=+1$. A fundamental Theorem by A. D. Aleksandrov
states that a Riemannian surface, homeomorphic to the sphere, with constant
curvature $K=+1$ and conical singularities of order $\beta_{i}\in(-1,0)$ can
be realized as the boundary of a convex polytope in the standard three-sphere
$\mathbb{S}^{3}$ (see [1, Chapter XII, p. 400] ). Note that from the previous
Theorem, a spherical polyhedron cannot have exactly one singularity.

The last result classifies the divisors that can be represented by a spherical
polyhedron having at least three singularities:

######  Theorem 6.2.

Let $\boldsymbol{\beta}=\sum_{i=1}^{n}\beta_{i}p_{i}$ be a divisor on
$S^{2}=\mathbb{C}\cup\infty$ such $n\geq 3$ and that $-1<\beta_{i}<0$ for all
$i$. Then there exists a unique conformal metric $g$ with constant curvature
$K=+1$ representing $\boldsymbol{\beta}$ if and only if

$0<2+\sum_{i=1}^{n}\beta_{i}<2(1+\min_{i}\\{\beta_{i}\\}).$ (4)

Expressed with the cone angles $\theta_{i}=2\pi(1+\beta_{i})$, the hypothesis
can also be written as $\theta_{i}<2\pi$ and

$0<4\pi+\sum_{i=1}^{n}(\theta_{i}-2\pi)<2\min_{i}\\{\theta_{i}\\}.$ (5)

The first inequality in this condition is none other than the Gauss-Bonnet
formula.

Let us observe that the condition (5) is similar to the condition satisfied by
the angles $\varphi_{1},\dots,\varphi_{n}$ of a spherical convex polygon:

$0<2\pi+\sum_{i=1}^{n}(\varphi_{i}-\pi)<2\min_{i}\\{\varphi_{i}\\}.$

The existence of a spherical metric representing $\boldsymbol{\beta}$ follows
from Theorem 5.1 above (see also [17]). The necessity of (4), as well as the
uniqueness of the metric have been proved by Feng Luo and Gang Tian in [8].

Note added in 2021. The subject of this survey has grown considerably over the
past 30 years. A nice reference covering more recent aspects of the theory is
the article [7] by M. Lai

## References

  * [1] A. D. Alexandrov selected works. Part II. Intrinsic geometry of convex surfaces. Edited by S. S. Kutateladze. Translated from the Russian by S. Vakhrameyev. Chapman and Hall/CRC, Boca Raton, FL, 2006.
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## Index

  * Cohn-Vossen inequality Proposition 3.3
  * Conical Singularity §1
  * Cusp §1
  * Divisor §2
  * Gauss-Bonnet Formula Theorem 3.1
  * Riemann-Hurwitz Formula Proposition 3.2
  * Simple singularity §1
  * Spherical polyhedra §6
  * The Berger–Nirenberg Problem §5
  * Uniformization Theorem (for the 2-sphere) Theorem 4.2


# Sequential Late Fusion Technique for Multi-modal Sentiment Analysis

Debapriya Banerjee  0000-0001-6666-5863 The University of Texas at
ArlingtonArlingtonTexasUSA76019<EMAIL_ADDRESS>,  Fotios
Lygerakis  0000-0001-8044-3511 The University of Texas at
ArlingtonArlingtonTexasUSA<EMAIL_ADDRESS>and  Fillia Makedon
The University of Texas at ArlingtonArlingtonTexasUSA<EMAIL_ADDRESS>
(2021)

###### Abstract.

Multi-modal sentiment analysis plays an important role for providing better
interactive experiences to users. Each modality in multi-modal data can
provide different viewpoints or reveal unique aspects of a user’s emotional
state. In this work, we use text, audio and visual modalities from MOSI
dataset and we propose a novel fusion technique using a multi-head attention
LSTM network. Finally, we perform a classification task and evaluate its
performance.

late fusion, multi-modal sentiment analysis, multi head attention recurrent
neural networks

††journalyear: 2021††copyright: acmcopyright††conference: The 14th PErvasive
Technologies Related to Assistive Environments Conference; June 29-July 2,
2021; Corfu, Greece††booktitle: The 14th PErvasive Technologies Related to
Assistive Environments Conference (PETRA 2021), June 29-July 2, 2021, Corfu,
Greece††price: 15.00††doi: 10.1145/3453892.3461009††isbn:
978-1-4503-8792-7/21/06††ccs: Computing methodologies Modeling methodologies

##  1\. Introduction

Social media is an industry that relies on efficient emotion recognition and
sentiment analysis (SA), in order to provide relevant content to the users by
exploiting their preferences and habits (Balazs and Velásquez, 2016).
Furthermore, the continuously increasing popularity of social robots is
another eminent field that can be greatly benefited by tracking user’s mental
state, offering more organic and engaging human-robot interaction (Shen et
al., 2015) or user-specific services (Scoglio et al., 2019).

##  2\. Related Work

The ever-growing user-provided content on social media consisting of different
modalities, i.e., audio, text and video, has already created huge dataset
(Zadeh et al., 2016), that eased the utilization of neural network (NN)
techniques on these fields. Literature has shown that multi-head attention
recurrent NN (MHA-RNN) in combination with late fusion (LF) (Stappen et al.,
2021; Kim and Lee, 2020) approaches are the most prominent techniques. As
discussed in (Majumder et al., 2018) by N.Majumder et al., sequential (or
hierarchical in their case) late fusion can filter out inter-modal
correlation.

In most recent approaches, an off-the-shelf encoder is used for each modality.
Then the encoding of each modality is being fused by an RNN. In (Kim and Lee,
2020), the authors use word2vec (Mikolov et al., 2013), OpenSmile (Eyben et
al., 2010) and 3D Convolutional NN (CNN) (Tran et al., 2015) for text, audio
and visual modalities respectively, the encoding of which they use in a
bidirectional MHA-RNN and a softmax layer at the end to perform binary
sentiment classification. A similar approach is followed by Poria et al. in
(Poria et al., 2018), but with the exception of using a custom CNN for the
text encoding and a support vector machine (SVM) for the classification task.

In this paper, we propose a LF technique that uses the same uni-modal encoders
as in (Kim and Lee, 2020). Our technique is also based on an MHA-RNN model,
however, the LF technique we use takes advantage of the sequential temporal
dependencies across the modality encodings.

##  3\. Method

###  3.1. Uni-modal Feature Extraction

In this work, we use LF combining uni-modal features extracted from 2199 short
video utterances of the MOSI dataset (Zadeh et al., 2016). The utterances were
annotated with sentiment labels ranging from -3 (very negative sentiment) to
+3 (very positive sentiment) and were produced by 89 speakers between 20-30
years old. From the video dataset, we first extract the audio and then convert
it into text using IBM Watson, a Speech to Text converter 111https://speech-
to-text-demo.ng.bluemix.net. Each modality consists of sequential utterances
that are fed into three different modality-specific encoders to extract the
visual, audio and textual uni-modal features. For textual features we use the
text-CNN model (Mikolov et al., 2013), for audio features we use OpenSmile
(Eyben et al., 2010) and for the visual ones 3D-CNN model (Tran et al., 2015).
All the above uni-modal extractors provide 300-dimensional features.

###  3.2. Fusion Technique

Our LF approach consists of two layers of LSTM nodes with multi-head attention
(MHA-LSTM). The first layer (LSTM Block in Figure 1) contains three nodes, one
for each of the three modalities, i.e. each node takes as input a sequence of
utterances of a particular modality from textual, audio or visual features. We
fuse the states of the first layer LSTM nodes in two steps:

1\. First, we fuse the states of the LSTM nodes that correspond to the textual
and audio features by calculating their inner product.

2\. Secondly, we calculate the inner product of the above outcome with the
state of the LSTM node that corresponds to the visual features.

We observed that fusing textual and audio features first gives better
performance. Consequently, the product of the second step is given as input to
the second layer of our MHA-LSTM network (LSTM in Figure 1). Finally, the
output of the MHA-LSTM network is given to a softmax layer which classifies
among the 7 different review categories of the dataset.

Figure 1. Fusion architecture using MHA-LSTM

###  3.3. Experiments

We train the MHA-LSTM network using the categorical cross-entropy as a loss
function and we optimize using ADAM. Our model converges after 100 epochs. We
split the dataset into 80% of training samples and 20% of testing ones. We
evaluate our model in terms of accuracy, precision, recall and F1 score. Table
1 shows the detailed evaluation metrics.

Table 1. Performance metrics on our proposed model with MOSI dataset Metric | Score
---|---
Accuracy | 0.769
Precision | 0.837
Recall | 0.781
F1 Score | 0.808

##  4\. Discussion and Future Work

In this work, we propose a novel multi-modal SA method using LF with an MHA-
LSTM model on uni-modal off-the-shelf feature extraction techniques. Our
intention is to explore the potential of a different LF technique using a
multi-head attention architecture. In the future, we plan to perform more
extensive experiments with other datasets on SA exploiting again multiple
modalities. These datasets will also involve other areas of sentiment like
Post-Traumatic Stress Disorder (PTSD) and general stress.

## References

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