# Non-positively curved Ricci Surfaces with catenoidal ends

Yiming ZANG  Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
<EMAIL_ADDRESS>
###### Abstract.

A Ricci surface is defined as a Riemannian surface $(M,g_{M})$ whose Gauss
curvature satisfies the differential equation $K\Delta
K+g_{M}\big{(}dK,dK\big{)}+4K^{3}=0$. Andrei Moroianu and Sergiu Moroianu
proved that a Ricci surface with non-positive Gauss curvature admits locally a
minimal immersion into $\mathbb{R}^{3}$. In this paper, we are interested in
studying non-compact orientable Ricci surfaces with catenoidal ends. We use an
analogue of the Weierstrass data to obtain some classification results for
such Ricci surfaces. We also give an existence result for positive genus Ricci
surfaces with catenoidal ends.

††MSC2021: 53A10, 53C20.††Keywords: Ricci surfaces; minimal surfaces;
catenoidal ends; Weierstrass data.

##  1\. Introduction

A classical question in the theory of minimal surfaces is to study when a
Riemannian surface $(M,g_{M})$ can be locally isometrically immersed into
$\mathbb{R}^{3}$ as a minimal surface. In the history, the first result for
this question was given by Gregorio Ricci-Curbastro. He provided a necessary
and sufficient condition for the existence of such immersions near points with
negative Gauss curvature (see (Ricci-Curbastro,, 1895)). Andrei Moroianu and
Sergiu Moroianu have proven later in (Moroianu and Moroianu,, 2015) that the
assumption of negative Gauss curvature could actually be left out. The main
idea of their proof is to study the differential equation

$\displaystyle K\Delta K+g_{M}\big{(}dK,dK\big{)}+4K^{3}=0$

satisfied by the Gauss curvature appearing in Ricci’s theorem. It is often
called the Ricci condition in the case $K<0$. A Riemannian surface whose
metric satisfies this equation is called a Ricci surface.

The theory of compact Ricci surfaces is developed by Andrei Moroianu and
Sergiu Moroianu as well. To be more precise, they provided some methods in
constructing compact Ricci surfaces. We are therefore interested in
considering non-compact orientable Ricci surfaces. In 1958, Huber proved that
a complete non-positively curved Riemannian surfaces with finite total
curvature has to be biholomorphic to a compact Riemann surface with finite
number of punctured points (see (Huber,, 1958)). Consequently, a problem which
should be naturally posed is to determine all the possible Ricci metrics on a
given Riemann surface. However, this project seems to be very ambitious
because many cases may occur around punctured points. The goal of our work is
to classify Ricci surfaces by adding an assumption on the punctured points
which is called the condition for catenoidal ends.

In this paper, Section 2 is devoted to the recall of some basic properties of
the well-known Weierstrass representation for minimal immersions in
$\mathbb{R}^{3}$. The definition of Ricci surfaces is introduced in Section 3,
then we focus on defining the Weierstrass data for Ricci surfaces which will
be our principal tools. In Section 4, we plan to define catenoidal ends for
Ricci surfaces, and we will prove a lemma providing a local description of the
Weierstrass data near a catenoidal end.

Our first main result is theorem 5.2 which will be proven in Section 5. This
theorem offers a complete classification of non-positively curved Ricci
surfaces $M\simeq\mathbb{C}\setminus\\{0\\}$ with two catenoidal ends. It
assures that a such Ricci surface can only be isometric to one of the two
classes of surfaces. We will give also an explanation for the relations
between our theorem and some related results. For example, theorem 5.2 is
equivalent to a result of Troyanov which classified all the metrics of
constant Gauss curvature 1 on $\bar{\mathbb{C}}$ with two conical
singularities at $0$ and $\infty$ (see (Troyanov,, 1989)).

In Section 6, we are going to discuss the case when the Ricci surface is
biholomorphic to $M\simeq\mathbb{C}\setminus\\{0,1\\}$ possessing three
catenoidal ends. Since things become much more complicated in this situation,
it could be very hard to obtain a complete classification as in the previous
case. We restrict ourselves to several classification results under some
additional conditions on the total curvature and the reducibility of induced
metric $(-K)g_{M}$. For concrete statements of these results, readers may
consult theorem 6.1, theorem 6.3 and theorem 6.4.

In the last section of this paper, we extend our discussion to Ricci surfaces
with positive genus. We have proven theorem 7.4 which assures that for
$k,n>0$, there exists a Ricci surface with genus $k$ and $n$ catenoidal ends.

The author wishes to express his gratitude to his supervisor, Prof. Benoît
Daniel, for his helpful discussions and encouragement.

##  2\. Preliminaries

In this section, we plan to recall the Weierstrass representation for minimal
immersions in $\mathbb{R}^{3}$ which plays an essential role in our
discussion.

Suppose that $X\mathrel{\mathop{\mathchar
58\relax}}M\rightarrow\mathbb{R}^{3}$ is a minimal immersion, where $M$ is a
smooth manifold of dimension 2. Taking a simply connected neighbourhood
$(U,(x,y))$ of $p_{0}\in M$ with $(x,y)$ the isothermal coordinates, then $X$
being minimal means that $X$ is a harmonic map, or equivalently,

(2.1) $\phi=(\phi_{1},\phi_{2},\phi_{3})\mathrel{\mathop{\mathchar
58\relax}}=\frac{\partial X}{\partial x}-i\frac{\partial X}{\partial
y}=2\frac{\partial X}{\partial z}$

is holomorphic, where $\frac{\partial}{\partial
z}=\frac{1}{2}(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})$ and
$\frac{\partial}{\partial\bar{z}}=\frac{1}{2}(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y})$. It is not difficult to verify that $\phi
dz$ is a globally defined vector valued holomorphic 1-form on $M$. With the
help of $\phi dz$, we may construct a meromorphic function $g$ and a
holomorphic 1-form $\eta$ on $M$ such that the minimal immersion $X$ can be
expressed as

(2.2)
$X(p)=X(p_{0})+\textrm{Re}\int_{p_{0}}^{p}\left(\frac{1}{2}\left(1-g^{2}\right)\eta,\frac{i}{2}\left(1+g^{2}\right)\eta,g\eta\right).$

Formula (2.2) is called the Weierstrass representation of the minimal
immersion $X\mathrel{\mathop{\mathchar 58\relax}}M\rightarrow\mathbb{R}^{3}$
and the pair $(g,\eta)$ is called the Weierstrass data of $X$. Since $\phi dz$
is a holomorphic 1-form which does not vanish, we see that whenever $g$ has a
pole of order $m$ at $p\in M$, $\eta$ must have a zero of order $2m$ at $p\in
M$. The function $g$ is also called the Gauss map. To see this, it is
sufficient to realize that $g$ is just the classical Gauss map
$G\mathrel{\mathop{\mathchar 58\relax}}M\rightarrow\mathbb{S}^{2}$ composed by
the stereographic projection from the north pole of $\mathbb{S}^{2}$.
Moreover, it is worth noticing that all these formulas still hold if we
multiply $\eta$ by an element $e^{i\theta}\in\mathbb{S}^{1}$, thus the new
Weierstrass data $(g,e^{i\theta}\eta)$ give another minimal immersion
$X_{\theta}\mathrel{\mathop{\mathchar 58\relax}}M\rightarrow\mathbb{R}^{3}$.
This leads to the following definition:

######  Definition 2.1.

The immersions $X_{\theta}\mathrel{\mathop{\mathchar
58\relax}}M\rightarrow\mathbb{R}^{3}$ with
$\theta\in\mathbb{R}/2\pi\mathbb{Z}$ are called associate minimal immersions
of $X\mathrel{\mathop{\mathchar 58\relax}}M\rightarrow\mathbb{R}^{3}$.

We will discuss some geometric data of a minimal immersion with the help of
the Weierstrass representation.

Firstly, if we denote locally $\eta$ by $\alpha dz$, then the induced
Riemannian metric $g_{M}$ on $M$ will be determined to be

(2.3) $\displaystyle
g_{M}=\frac{1}{4}|\alpha|^{2}\big{(}1+|g|^{2}\big{)}^{2}|dz|^{2}.$

The second thing is to determine the Gauss curvature $K$ of $M$. In use of the
Riemannian metric $g_{M}$, the Gauss curvature can be computed as

(2.4) $\displaystyle
K=-\left[\frac{4|g^{\prime}|}{|\alpha|(1+|g|^{2})^{2}}\right]^{2}.$

We list in the end some examples of minimal immersions in $\mathbb{R}^{3}$
with their Weierstrass data:

######  Example 2.2.

(Enneper’s Surface) Let $M=\mathbb{C}$, the Weierstrass data are given by
$g(z)=z$ and $\eta=dz$.

######  Example 2.3.

(Catenoid) Let $M=\mathbb{C}\backslash\\{0\\}$, the Weierstrass data are given
by $g(z)=z$ and $\eta=\frac{a}{z^{2}}dz$, with
$a\in\mathbb{R}\backslash\\{0\\}$.

##  3\. Ricci metrics and Ricci surfaces

Andrei Moroianu and Sergiu Moroianu defined in their article Moroianu and
Moroianu, (2015) a class of surfaces called Ricci surfaces which can be
regarded as a generalisation of minimal surfaces in $\mathbb{R}^{3}$. In this
section, we are going to introduce some basic properties of Ricci surfaces.
Throughout this paper, we only consider orientable Ricci surfaces.

The following theorem is an important fact which has been proven by Gregorio
Ricci-Curbastro Ricci-Curbastro, (1895):

######  Theorem 3.1.

A Riemannian surface $(M,g_{M})$ with negative Gauss curvature $K<0$ has local
isometric immersions as minimal surface in $\mathbb{R}^{3}$ if and only if one
of the three equivalent conditions holds:
i). The metric $\sqrt{-K}g_{M}$ is a flat metric;
ii). The metric $(-K)g_{M}$ is a metric of constant Gauss curvature $1$;
iii). The Gauss curvature satisfies

$\displaystyle K\Delta K+g_{M}\big{(}dK,dK\big{)}+4K^{3}=0.$

The third condition in theorem 3.1 can even be studied without the hypothesis
$K<0$. This observation inspires Andrei Moroianu and Sergiu Moroianu to give
the following definition.

######  Definition 3.2.

A Riemannian surface $(M,g_{M})$ whose Gauss curvature $K$ satisfies the
following identity

(3.1) $\displaystyle K\Delta K+g_{M}\big{(}dK,dK\big{)}+4K^{3}=0$

is called a Ricci surface, the metric $g_{M}$ is called a Ricci metric.

From theorem 3.1, we can see that all the immersed minimal surfaces in
$\mathbb{R}^{3}$ are Ricci surfaces. Moreover, Andrei Moroianu and Sergiu
Moroianu proved in their article the following result which shows that a Ricci
surface with non-positive Gauss curvature can be locally realized as an
immersed minimal surface in $\mathbb{R}^{3}$.

######  Theorem 3.3.

Let $(M,g_{M})$ be a connected Ricci surface, then its Gauss curvature $K$
does not change sign on $M$. If $K\leq 0$, then $M$ admits locally a minimal
isometric immersion into $\mathbb{R}^{3}$.

These two theorems indicate that the relation between Ricci surfaces with non-
positive Gauss curvature and minimal immersions in $\mathbb{R}^{3}$ is very
close. Therefore, we are inspired to use the tools in the theory of minimal
immersions in $\mathbb{R}^{3}$ to study Ricci surfaces. This is possible
thanks to the next proposition.

######  Proposition 3.4.

Let $g_{0}=|dz|^{2}$ be a flat metric on some domain $\Omega\subset\mathbb{C}$
and $f\mathrel{\mathop{\mathchar 58\relax}}\Omega\rightarrow\mathbb{R}$ a
smooth function. The metric $g_{\Omega}=e^{-2f}g_{0}$ admits locally a minimal
immersion in $\mathbb{R}^{3}$ if and only if near every $x\in\Omega$ there
exists a pair of holomorphic functions $(a,b)$ such that
$e^{-f}=|a|^{2}+|b|^{2}$.

###### Proof.

(1). $\Rightarrow$ If a such isometric immersion exists, then from the
discussion in Section 2, there are Weierstrass data $\eta=\alpha dz$ and $g$
satisfying $e^{-f}=\frac{1}{2}|\alpha|(1+|g|^{2})$. Since $\alpha$ can only
have a zero of order $2m$ where $g$ has a pole of order $m$, we may find two
holomorphic functions $a$ and $b$ such that

$\displaystyle a^{2}=\frac{\alpha}{2},\quad b=ag,$

thus $e^{-f}=|a|^{2}+|b|^{2}$.

(2). $\Leftarrow$ If there are two holomorphic functions $a$ et $b$ such that
$e^{-f}=|a|^{2}+|b|^{2}$, then $a$ can be supposed to be non-vanishing on a
disc $D$ centered at $x$. Let us define $\alpha\mathrel{\mathop{\mathchar
58\relax}}=2a^{2}$, $g\mathrel{\mathop{\mathchar 58\relax}}=\frac{b}{a}$, then
$\eta\mathrel{\mathop{\mathchar 58\relax}}=\alpha dz$ is a holomorphic 1-form
and $g$ is a meromorphic function. Additionally, zeroes of $\eta$ are
compatible with the poles of $g$ as mentioned in Section 2. Taking $(g,\eta)$
as Weierstrass data, we may find an isometric minimal immersion into
$\mathbb{R}^{3}$ whose metric is $g_{\Omega}$. ∎

Moreover, Calabi has proven the following theorem of rigidity (see Calabi,
(1968), Daniel, (2021)):

######  Theorem 3.5.

Two minimal isometric immersions from a simply connected surface $M$ into
$\mathbb{R}^{3}$ are associate, up to the action of $Isom(\mathbb{R}^{3})$ on
$\mathbb{R}^{3}$.

######  Remark 3.6.

This theorem tells us that the minimal isometric immersion from a simply
connected surface $M$ into $\mathbb{R}^{3}$ is unique up to the action of
$Isom(\mathbb{R}^{3})$ and the $\mathbb{S}^{1}$-action (associate minimal
immersions). Combining proposition 3.4 and Calabi’s theorem 3.5, we may
locally identify the non-positively curved Ricci surface with the equivalent
class of its minimal isometric immersions into $\mathbb{R}^{3}$. This
identification will be of great significance in our further discussion.

We are now interested in studying how the identification mentioned above
allows us to do with the Weierstrass data. Firstly, it is easy to be seen from
(2.2) that the Weierstrass data remain invariant under translations in
$\mathbb{R}^{3}$. Secondly, definition 2.1 implies that we can multiply $\eta$
by an element $e^{i\theta}\in\mathbb{S}^{1}$ while keeping $g$ fixed. Thirdly,
let us make a rotation in $\mathbb{R}^{3}$, i.e., applying an element $A\in
SO(3,\mathbb{R})$ to Weierstrass representation (2.2), then the new surface
$\tilde{X}$ and its Gauss map $\tilde{G}$ are written as

(3.2) $\displaystyle\tilde{X}=AX,\quad\tilde{G}=AG.$

Owing to the fact that $\frac{\partial f}{\partial
z}=2\frac{\partial\textrm{Re}f}{\partial z}$ for every holomorphic function
$f$, we obtain

(3.3) $\displaystyle\tilde{\phi}\mathrel{\mathop{\mathchar
58\relax}}=2\frac{\partial\tilde{X}}{\partial z}=A\phi.$

It is known that the stereographic projection $\pi$ and rotations preserve
angles, so the composite map

$\displaystyle\pi\circ A\circ\pi^{-1}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{C}}\rightarrow\bar{\mathbb{C}}$

is conformal, hence an element of $Aut(\bar{\mathbb{C}})=PSL(2,\mathbb{C})$.
In order to find the new Weierstrass data, we need the following classical
result whose proof is standard.

######  Lemma 3.7.

The map

$\displaystyle\varphi\mathrel{\mathop{\mathchar 58\relax}}SO(3,\mathbb{R})$
$\displaystyle\rightarrow PSL(2,\mathbb{C})$ $\displaystyle A$
$\displaystyle\mapsto\pi\circ A\circ\pi^{-1}$

is an injective Lie group homomorphism with $\textrm{Im}\varphi=PSU(2)$, hence
it induces an isomorphism $SO(3,\mathbb{R})\simeq PSU(2)$.

Now we are going to determine the new Weierstrass data after the rotation. As
we have already seen, $g=\pi\circ G\mathrel{\mathop{\mathchar
58\relax}}M\rightarrow\bar{\mathbb{C}}$ is the Gauss map, hence

(3.4) $\displaystyle\tilde{g}=\pi\circ A\circ G=\pi\circ
A\circ\pi^{-1}\circ\pi\circ G=\varphi(A).g,$

where $\varphi(A).g$ denotes the action of $\varphi(A)$ on $g$. Then by
combining (2.2), (3.3) and (3.4), we may get the new Weierstrass data
$(\tilde{g},\tilde{\eta})$.

Let us consider the Hopf differential defined by $Q\mathrel{\mathop{\mathchar
58\relax}}=dg\otimes\eta$. It is a direct observation from the previous
calculation that $Q$ is invariant under rotations in $\mathbb{R}^{3}$.
Therefore, the new Weierstrass data obtained by applying $A\in
SO(3,\mathbb{R})$ are computed as

(3.5) $\left\\{\begin{aligned} &~{}\tilde{g}=\varphi(A).(g),\\\
&~{}\tilde{\eta}=\frac{Q}{d\tilde{g}}.\\\ \end{aligned}\right.$

From the proof of proposition 3.4, a non-positively curved Ricci surface can
be locally equipped with the Weierstrass data. It is worthwhile noticing that
the Weierstrass data may not be globally defined on Ricci surfaces in general.
However, they are always well-defined on the universal cover of a Ricci
surface. In fact, for a non-positively curved Ricci surface $(M,g_{M})$, the
metric defined by $g_{1}\mathrel{\mathop{\mathchar 58\relax}}=(-K)g_{M}$ is a
metric of constant Gauss curvature 1, possibly with isolated conical
singularities at zeroes of $K$ (see (Moroianu and Moroianu,, 2015), Troyanov,
(1989)). It induces on the universal cover $\tilde{M}$ of $M$ a metric
$d\sigma^{2}$ of constant Gauss curvature 1. Then there exists a meromorphic
function $g\mathrel{\mathop{\mathchar
58\relax}}\tilde{M}\rightarrow\bar{\mathbb{C}}$ such that

(3.6) $\displaystyle d\sigma^{2}=\frac{4dgd\bar{g}}{(1+|g|^{2})^{2}}.$

Moreover, this function $g$ is unique up to an action of $PSU(2)$ as in (3.4).
Let us take $g$ as the Gauss map, then the Weierstrass data can be constructed
with the help of proposition 3.4 and theorem 3.5. To be convenient, we still
call it the Weierstrass data of this Ricci surface. A point that should be
mentioned is that the metric $d\sigma^{2}$ is invariant under the action of
the deck transformation group $Deck(\tilde{M}/M)$. It is immediate from the
uniqueness of $g$ that every deck transformation of $\tilde{M}$ corresponds to
an element in $PSU(2)$. More precisely, if we identify the deck transformation
group $Deck(\tilde{M}/M)$ with the fundamental group $\pi_{1}(M)$ of $M$, then
there exists a monodromy representation $\rho\mathrel{\mathop{\mathchar
58\relax}}\pi_{1}(M)\rightarrow PSU(2)$ such that

(3.7) $\displaystyle g\circ\tau^{-1}=\rho(\tau).g,$

for all $\tau\in\pi_{1}(M)$. This monodromy representation will be very
important in our further discussion.

Even though the Hopf differential $Q$ is only defined on the universal cover
$\tilde{M}$ in general, its modulus $|Q|$ is actually well-defined on the
Ricci surface $M$ itself. To see this, since $Q$ is invariant under the action
of $PSU(2)$ and it may differ just by a possible multiplication of
$e^{i\theta}\in\mathbb{S}^{1}$, the modulus $|Q|$ is therefore independant of
all these actions. Hence it descends to a well-defined quantity on $M$.

Thanks to the identification mentioned in remark 3.6, all the Weierstrass data
under these operations are regarded to be equivalent. This equivalence class
of Weierstrass data is exactly what the Weierstrass data of the non-positively
curved Ricci surface means. From now on, once a representative pair of
Weierstrass data is fixed, we are allowed to apply all the operations
discussed above.

To finish this section, we also give an example of Ricci surfaces. For CMC-1
immersions in the hyperbolic 3-space $\mathcal{H}^{3}$, we may define an
analogue of the Weierstrass representation (see (Bryant,, 1987), Umehara and
Yamada, (1993)). We know from the Lawson’s correspondence (see (H. Blaine
Lawson,, 1970)) that CMC-1 immersions in $\mathcal{H}^{3}$ are locally
isometric to minimal immersions in $\mathbb{R}^{3}$, hence it provides some
Ricci surfaces. A typical example is the following:

######  Example 3.8.

(Catenoid cousins) Let $M=\mathbb{C}\backslash\\{0\\}$, the representative
Weierstrass data are $g(z)=z^{\mu}$ and $\eta=az^{-\mu-1}dz$, with
$a,\mu\in\mathbb{R}_{+}^{*}$ and $\mu\neq 1$. In particular, if $\mu$ is an
integer, then it is isometric to a $\mu$-fold cover of catenoid.

##  4\. Definition of catenoidal ends

In this section we plan to define catenoidal ends for a non-positively curved
Ricci surface with the help of the Weierstrass data. Before beginning, we need
some preparation.

######  Definition 4.1.

Let $(M,g_{M})$ be a complete surface, then the integral

(4.1) $\displaystyle K(M)\mathrel{\mathop{\mathchar 58\relax}}=\int_{M}KdA$

is called the total curvature of $M$, where $dA$ is the area element and $K$
is the Gauss curvature.

It is well-know that every two-dimensional orientable smooth Riemannian
manifold can be equipped with a compatible complex structure so that it may
also be regarded as a Riemann surface. We announce here an important theorem
proved by Huber Huber, (1958) and Osserman Osserman, (2013):

######  Theorem 4.2.

Let $(M,g_{M})$ be a complete non-positively curved orientable Riemannian
surface with finite total curvature. Then there exists a compact Riemann
surface $S_{k}$ of genus $k$ and a finite number of points $p_{1},...,p_{r}$
on $S_{k}$ such that $M$ is biholomorphic to
$S_{k}\backslash\\{p_{1},...,p_{r}\\}$.

Theorem 4.2 permits us to identify a complete non-positively curved Ricci
surface which has finite total curvature with a Riemann surface
$S_{k}\backslash\\{p_{1},...,p_{r}\\}$. It is natural for us to study the
Ricci surface with some particular assumptions at the punctured points.

Example 2.3 and 3.8 showed some complete non-positively curved Ricci surfaces
which are biholomorphic to $\bar{\mathbb{C}}\setminus\\{0,\infty\\}$. Their
universal covers are isometric to that of a classical catenoid. In addition,
the Hopf differential $Q$ of these examples is meromorphic near each punctured
points. We have seen in the previous section that the Hopf differential $Q$ is
independent of isometries of $\mathbb{R}^{3}$, and it is unique up to a
multiplication by an element $e^{i\theta}\in\mathbb{S}^{1}$. Hence we are
inspired to study the order of $Q$ at punctured points. At the punctured point
$0$, it is not difficult to observe that $\textrm{ord}_{0}\ Q=-2$. Similarly,
at the punctured point $\infty$, we replace $z$ by $\frac{1}{w}$, then an easy
calculation shows that $\textrm{ord}_{\infty}\ Q=-2$ still holds. The
condition $\textrm{ord}\ Q=-2$ is therefore valuable for us. Moreover, it is
worth remarking that $\textrm{ord}\ Q=-2$ does not depend on the choice of
conformal parameters. For these reasons, we may give the following definition.

######  Definition 4.3.

A catenoidal end of a non-positively curved Ricci surface $M\simeq
S_{k}\setminus\\{p_{1},p_{2},...,p_{r}\\}$ with finite total curvature is a
punctured point $p_{i}$ which possesses a neighborhood where the Hopf
differential $Q$ is well-defined and meromorphic, satisfying
$\textrm{ord}_{p_{i}}\ Q=-2$.

For a Ricci surface
$M\simeq\bar{\mathbb{C}}\setminus\\{p_{1},p_{2},...,p_{r}\\}$ with $r$
catenoidal ends, the Hopf differential $Q$ is a well-defined holomorphic
quadratic differential on $M$. In fact, the fundamental group $\pi_{1}(M)$ is
generated by closed curves $\gamma_{1},...,\gamma_{r}$ surrounding each
punctured point. By definition 4.3, $Q$ is independent of all the generators
of $\pi_{1}(M)$, thus well-defined on $M$. However, this property fails to
hold for Ricci surfaces with positive genus in general. This is because there
will be other generators of the fundamental group $\pi_{1}(M)$ than the closed
curves mentioned above, hence $Q$ may not be well-defined on $M$ itself.

Taking advantage of (2.3) and (2.4), we may observe that the modulus $|Q|$ of
$Q$ coincides with the metric $\sqrt{-K}g_{M}$ defined in theorem 3.1.
Therefore, around a catenoidal end which corresponds to $z=0$ of a Ricci
surface $(M,g_{M})$, the induced flat metric $\sqrt{-K}g_{M}$ can be locally
written as

(4.2)
$\displaystyle\sqrt{-K}g_{M}=\left(\frac{a}{|z|^{2}}+o\left(\frac{1}{|z|^{2}}\right)\right)\left|dz\right|^{2},$

with $a\in\mathbb{R}\setminus\\{0\\}$. Actually, the consverse is also true.
To see this, let us assume that the induced metric $\sqrt{-K}g_{M}$ of a non-
positively curved Ricci surface has a local expression as in (4.2) near a
punctured point $z=0$. If we consider a closed curve $\gamma$ surrounding this
point, then by Calabi’s rigidity theorem 3.5 and the definition of $Q$, we
must have

$\displaystyle Q\circ\tau^{-1}=e^{2\pi i\theta}Q$

for some $\theta\in[0,1)$, where $\tau=[\gamma]\in\pi_{1}(M)$. It can be
checked that $P\mathrel{\mathop{\mathchar 58\relax}}=z^{-\theta}Q$ is a well-
defined meromorphic quadratic differential near $0$. Moreover, the modulus of
$P$ takes the form

$\displaystyle|P|=|z|^{-\theta}|Q|=\left(\frac{a}{|z|^{2+\theta}}+o\left(\frac{1}{|z|^{2+\theta}}\right)\right)\left|dz\right|^{2},$

it implies that $0$ is a pole of $P$ and $\theta=0$. Hence $Q$ is locally
well-defined near $0$ and $z=0$ is a catenoidal end of the Ricci surface $M$.
Consequently, the catenoidal end of a non-positively curved Ricci surface may
be equivalently defined to be a punctured point where the induced flat metric
$\sqrt{-K}g_{M}$ has the expression as in (4.2).

Apart from the metric $\sqrt{-K}g_{M}$, we can say even more about the
Weierstrass data of the Ricci surface. The lemma below provides locally an
explicit description of the catenoidal end of Ricci surfaces.

######  Lemma 4.4.

For a non-positively curved Ricci surface $(M,g_{M})$ with finite total
curvature, it has a catenoidal end at the punctured point corresponding to
$z=0$ if and only if there exists an element $A\in SO(3,\mathbb{R})$ such that
after the rotation, the Weierstrass data can be locally written as

(4.3) $\left\\{\begin{aligned}
~{}\tilde{\eta}&=\tilde{\alpha}dz=\left(\frac{a}{z^{\lambda+1}}+o\left(\frac{1}{z^{\lambda+1}}\right)\right)dz,\\\
~{}\tilde{g}&=bz^{\lambda}+o(z^{\lambda}),\\\ \end{aligned}\right.$

where $a,b\in\mathbb{C}\backslash\\{0\\}$ and $\lambda\in\mathbb{R}^{*}_{+}$.

###### Proof.

$\Leftarrow$ If there is an element $A\in SO(3,\mathbb{R})$ such that after
the rotation, the Weierstrass data $\tilde{\eta}$ and $\tilde{g}$ have form of
(4.3), then it can be easily checked that $\textrm{ord}\ Q=-2$ is verified at
that point.

$\Rightarrow$ Conversely, given $\textrm{ord}\ Q=-2$ at the punctured point
$z=0$, we are obliged to consider an element $\tau=[\gamma]\in\pi_{1}(M)$,
where $\gamma$ is a closed curve around this punctured point. Since each
rotation of $\bar{\mathbb{C}}$ is conjugate to a rotation fixing $0$ and
$\infty$, in use of lemma 3.7, we may suppose that the function $g$ satisfies

$\displaystyle g\circ\tau^{-1}=e^{2\pi i\theta}g,$

with $\theta\in[0,1)$. One may verify that $h(z)\mathrel{\mathop{\mathchar
58\relax}}=z^{-\theta}g$ is a well-defined meromorphic function near $z=0$.
The condition that the total curvature of $M$ is finite tells us that $h$ is
actually meromorphic at $0$ (see (Bryant,, 1987), Prop 4), thus the Gauss map
$g$ can be locally written as
$g=z^{\mu}(\sum\limits_{j=0}^{\infty}{b_{j}z^{j}})$ with $\mu\in\mathbb{R}$,
$b_{j}\in\mathbb{C}$ and $b_{0}\neq 0$.

(1). Assume that $\mu>0$, then it is immediate from the definition of $Q$ that

$\left\\{\begin{aligned}
&~{}g=z^{\mu}\left(\sum\limits_{j=0}^{\infty}{b_{j}z^{j}}\right),\\\
&~{}\eta=z^{-\mu-1}\left(\sum\limits_{j=0}^{\infty}{a_{j}z^{j}}\right)dz,\\\
\end{aligned}\right.$

with $a_{0},b_{0}\in\mathbb{C}\backslash\\{0\\}$. This is the expression (4.3)
and we can just take $A=I$.

(2). Now suppose that $\mu\leq 0$.

(i). If $\mu=0$, then $g=b_{0}+\sum\limits_{j=1}^{\infty}{b_{j}z^{j}}$. The
condition $\textrm{ord}\ Q=-2$ assures that $g$ is not a constant. Thus we may
take a rotation $\tilde{g}=\frac{g-b_{0}}{\bar{b_{0}}g+1}$. Since in this case
$\tilde{g}$ is meromorphic near $z=0$ and $\tilde{g}(0)=0$, it must have the
form

$\tilde{g}=z^{n}\left(\sum\limits_{j=0}^{\infty}{c_{j}z^{j}}\right)$

for some $n\in\mathbb{N}^{*}$, where $c_{j}\in\mathbb{C}$ and $c_{0}\neq 0$.
The isomorphism constructed in lemma 3.7 implies immediately the existence of
such an element $A\in SO(3,\mathbb{R})$. In addition, the corresponding
$\tilde{\eta}$ can be proved to be

$\tilde{\eta}=z^{-n-1}\left(\sum\limits_{j=0}^{\infty}{a_{j}z^{j}}\right)dz,$

with $a_{j}\in\mathbb{C}$ and $a_{0}\neq 0$. This satisfies the expression
(4.3).

(ii). In the case $\mu<0$, we will apply the transformation
$\tilde{g}=-\frac{1}{g}$. It is easy to see that the new Weierstrass data have
the form as in (4.3). The corresponding $A\in SO(3,\mathbb{R})$ is also given
by lemma 3.7. This completes the proof. ∎

With the help of this lemma, the Ricci metric can be locally written as

(4.4) $\displaystyle
g_{M}=\left(\frac{b}{|z|^{2\lambda+2}}+O\left(\frac{1}{|z|^{2}}\right)\right)\left|dz\right|^{2},$

where $b\in\mathbb{R}\setminus\\{0\\}$ and $\lambda\in\mathbb{R}^{*}_{+}$.
Moreover, taking (4.2) and (4.4) into consideration, we obtain immediately
from Ricci’s theorem 3.1 a local expression for $(-K)g_{M}$ which is

(4.5)
$\displaystyle(-K)g_{M}=\left(c|z|^{2\lambda-2}+o\left(|z|^{2\lambda-2}\right)\right)\left|dz\right|^{2},$

with $c\in\mathbb{R}\setminus\\{0\\}$. Since $\lambda>0$, the metric
$(-K)g_{M}$ has a conical singularity of order $\lambda-1>-1$ at the
catenoidal end $z=0$.

It should be noticed that this lemma is just a local result. In order to do a
global discussion, we need to study the monodromy representation introduced in
the previous section. Inspired by Umehara and Yamada in their article Umehara
and Yamada, (2000), we will give the following definitions.

######  Definition 4.5.

Let $d\sigma^{2}$ be a metric with constant constant Gauss curvature 1 and
finite number of conical singularities on $M$,

i). We call it an irreducible metric if the image of the corresponding
monodromy representation $\rho\mathrel{\mathop{\mathchar
58\relax}}\pi_{1}(M)\rightarrow PSU(2)$ is not abelian;

ii). We call it a non-trivially reducible metric if the image of $\rho$ is
abelian but not trivial;

iii). It is called a trivially reducible metric if the image of $\rho$ is
trivial.

For a non-positively curved Ricci surface $(M,g_{M})$ with finite total
curvature and more than one ends, if the induced metric
$g_{1}\mathrel{\mathop{\mathchar 58\relax}}=(-K)g_{M}$ is reducible, then we
can apply lemma 4.4 one by one to all the catenoidal ends. Since the image of
the monodromy representation $\rho$ is abelian, the order of this procedure
does not influence the final result. Hence in this case, we may do some global
analysis. However, this method does not work if the induced metric $g_{1}$ is
irreducible.

##  5\. $M\simeq\mathbb{C}\setminus\\{0\\}$ with two catenoidal ends

We are inspired by Huber’s theorem to study non-positively curved Ricci
surfaces with catenoidal ends. It is immediate from the Riemann uniformization
theorem that the universal cover of a such Ricci surface can only be the whole
plane $\mathbb{C}$ or the upper half plane $\mathbb{H}$. In the $\mathbb{C}$
case, the corresponding Ricci surface $M$ is biholomorphic to either
$\mathbb{C}\setminus\\{0\\}$ or $\mathbb{C}$. The following theorem affirms
that the case when $M\simeq\mathbb{C}$ cannot appear.

######  Theorem 5.1.

There does not exist non-positively curved Ricci surface $M\simeq\mathbb{C}$
with exactly one catenoidal end.

###### Proof.

Let us identify $M$ with $\bar{\mathbb{C}}\setminus\\{0\\}$, then the
catenoidal end corresponds to $0$. Applying lemma 4.4, we may assume that the
Gauss map $g$ has a pole at $0$. Since $\pi_{1}(M)$ is trivial, $g$ can be
written as $g=z^{-n}\varphi(z)$, where $n\in\mathbb{N}^{*}$ and $\varphi$ is a
meromorphic function on $\bar{\mathbb{C}}$ satisfying $\varphi(0)\neq 0$. The
condition for catenoidal ends leads to the fact that $\textrm{ord}\
\eta=-2-(-n-1)=n-1\geq 0$ at $0$. Therefore, $\eta$ is a holomorphic 1-form on
$\bar{\mathbb{C}}$ thus $\eta\equiv 0$. This implies that $Q\equiv 0$, which
is not possible. ∎

Thanks to theorem 5.1, we may focus on giving a classification of non-
positively curved Ricci surfaces $M\simeq\mathbb{C}\setminus\\{0\\}$ with two
catenoidal ends.

######  Theorem 5.2.

A complete non-positively curved Ricci surface
$M\simeq\mathbb{C}\backslash\\{0\\}$ with two catenoidal ends can only be
isometric to a catenoid, a catenoid cousin given as in example 3.8 or the
surface determined by the Weierstrass data

(5.1) $g=z^{n}+a,\quad\quad\eta=bz^{-n-1}dz,$

where $a,b\in\mathbb{R}^{*}_{+}$ and $n\in\mathbb{N}^{*}$. Moreover, all these
surfaces are mutually non-isometric.

###### Proof.

On the one hand, since $Deck(\tilde{M}/M)\simeq\pi_{1}(M)\simeq\mathbb{Z}$ is
a cyclic group, the proof of lemma 4.4 tells us that
$h(z)\mathrel{\mathop{\mathchar 58\relax}}=z^{-\theta}g$ is a meromorphic
function on $\bar{\mathbb{C}}$ for some $\theta\in[0,1)$. Therefore, we may
assume that the Gauss map is $g(z)=z^{\mu}f(z)$, where $f$ is a meromorphic
function on $\bar{\mathbb{C}}$ and $\mu>0$. Hence it must have the form

(5.2) $\displaystyle g(z)=z^{\mu}\frac{\varphi(z)}{\psi(z)},$

where $\varphi$ and $\psi$ are coprime polynomials such that $\psi(0)\neq 0$
and $\varphi(0)\neq 0$. From lemma 4.4 and the fact that $\eta$ can only have
a zero of order $2m$ where $g$ has a pole of order $m$, we obtain immediately

(5.3) $\displaystyle\eta=\lambda z^{-\mu-1}\psi(z)^{2}dz,$

where $\lambda\in\mathbb{C}\backslash\\{0\\}$ is a constant number. On the
other hand, as we have explained in the previous section, the Hopf
differential $Q=dg\otimes\eta$ is meromorphic on $\bar{\mathbb{C}}$ satisfying
$\textrm{ord}\ Q=-2$ at $0$ and $\infty$, thus it can be written as

(5.4) $\displaystyle Q=\frac{\nu}{z^{2}}dz\otimes dz,$

where $\nu\in\mathbb{C}\backslash\\{0\\}$ is a constant number.

Taking (5.2), (5.3) and (5.4) into consideration, we have

$\displaystyle Q$ $\displaystyle=dg\otimes\eta$ $\displaystyle=\lambda
z^{-2}[\mu\varphi\psi+z(\varphi^{\prime}\psi-\varphi\psi^{\prime})]dz\otimes
dz$ $\displaystyle=\nu z^{-2}dz\otimes dz.$

Comparing these two expressions, we draw the following equation

(5.5)
$\displaystyle\mu\varphi\psi+z(\varphi^{\prime}\psi-\varphi\psi^{\prime})=\frac{\nu}{\lambda}.$

Differentiating the two sides of (5.5), it becomes

$\displaystyle\psi[z\varphi^{\prime\prime}+(1+\mu)\varphi^{\prime}]=\varphi[z\psi^{\prime\prime}+(1-\mu)\psi^{\prime}].$

Since $\varphi$ and $\psi$ are coprime, there must be a polynomial $h(z)$ such
that

(5.6) $\displaystyle
z\varphi^{\prime\prime}+(1+\mu)\varphi^{\prime}=h(z)\varphi.$

The degree of the left-hand side of (5.6) is at most $\textrm{deg}\
\varphi-1$, while the degree of the right-hand side should be at least
$\textrm{deg}\ \varphi$ unless $h(z)=0$. This comparison tells us that
$h(z)=0$. Therefore, (5.6) is actually

(5.7) $\displaystyle z\varphi^{\prime\prime}+(1+\mu)\varphi^{\prime}=0.$

By solving differential equation (5.7), we get

(5.8) $\displaystyle\varphi=a+bz^{-\mu},$

where $a,b\in\mathbb{C}$ are constant numbers.

Similarly, we can get

(5.9) $\displaystyle\psi=cz^{\mu}+d,$

where $c,d\in\mathbb{C}$ are constant numbers.

Since $\mu>0$ and $\varphi$ is a polynomial, we must have $b=0$.

(i). If $\mu\notin\mathbb{N}$, then $c=0$ because $\psi$ should also be a
polynomial. Thus the Weierstrass data of the Ricci surface are

$\displaystyle g=\frac{a}{d}z^{\mu},\quad\quad\eta=\lambda d^{2}z^{-\mu-1}dz.$

With a change of variables $u=(\frac{a}{d})^{\frac{1}{\mu}}z$, these
expressions can be simplified to be

$\displaystyle g=u^{\mu},\quad\quad\eta=\lambda adu^{-\mu-1}du.$

Moreover, by applying an $\mathbb{S}^{1}$-action if necessary, we may suppose
$\lambda ad$ to be a real number $\rho>0$, thus the Weierstrass data of the
Ricci surface can be written as

(5.10) $\displaystyle g=u^{\mu},\quad\quad\eta=\rho u^{-\mu-1}du$

with $\rho\in\mathbb{R}^{*}_{+}$.

(ii). If $\mu\in\mathbb{N}$ and $c=0$, then the Weierstrass data have exactly
the same form as in (5.10). In these two cases, the Ricci surface $M$ is
isometric to a catenoid if $\mu=1$, to a catenoid cousin if $\mu\neq 1$.

(iii). If $\mu\in\mathbb{N}$ and $c\neq 0$, the Weierstrass data will have the
form

(5.11) $\displaystyle
g=\frac{az^{\mu}}{cz^{\mu}+d},\quad\quad\eta=\lambda(cz^{\mu}+d)^{2}z^{-\mu-1}dz.$

Denoting

$\displaystyle r=\frac{a}{d}\left(\frac{|c|^{2}}{|a|^{2}}+1\right),\quad
s=\frac{\bar{c}}{\bar{a}},\quad t=\frac{\lambda
d^{2}}{\frac{|c|^{2}}{|a|^{2}}+1},$

then a simple computation shows that after a rotation, the Weierstrass data of
the Ricci surface become

(5.12)
$\displaystyle\tilde{g}=rz^{\mu}+s,\quad\quad\tilde{\eta}=tz^{-\mu-1}dz,$

where $r,s,t\in\mathbb{C}\backslash\\{0\\}$. With the help of a change of
variables $w=r^{-\frac{1}{\mu}}z$, (5.12) is simplified as

(5.13)
$\displaystyle\tilde{g}=w^{\mu}+s,\quad\quad\tilde{\eta}=trw^{-\mu-1}dw.$

Applying an $\mathbb{S}^{1}$-action if necessary, we may assume $tr$ to be a
positive real number $\rho$, hence the Weierstrass data can be written as

(5.14) $\displaystyle\tilde{g}=w^{\mu}+s,\quad\quad\tilde{\eta}=\rho
w^{-\mu-1}dw,$

where $s\in\mathbb{C}\backslash\\{0\\}$ and $\rho\in\mathbb{R}^{*}_{+}$.
Moreover, if we denote $s=\xi e^{i\theta}$ with $\xi\in\mathbb{R}^{*}_{+}$ and
$\theta\in\mathbb{R}/2\pi\mathbb{Z}$, then it will be immediate by a rotation
and a change of variables that the formula (5.1) is obtained.

From Ricci’s theorem 3.1, we know that a Ricci metric $g_{M}$ induces natually
a metric $(-K)g_{M}$ which is of constant Gauss curvature 1, possibly with
conical singularities. Therefore, we are able to compute the total curvature
with the help of Gauss-Bonnet formula with conical singularities

(5.15)
$\displaystyle\frac{1}{2\pi}\int_{M}(-K)dA=\chi(\bar{M})+\sum_{i=1}^{n}\beta_{i},$

where $\beta_{i}>-1$ are the orders of conical singularities (see (Troyanov,,
1989)). Taking advantage of the two expressions (5.10) and (5.1), a direct
computation shows that the metric $(-K)g_{M}$ has two conical singularities of
the same order at $0$ and $\infty$, which is $\mu-1$ for (5.10) and $n-1$ for
(5.1). Hence the total curvature is $-4\pi\mu$ for (5.10) and $-4\pi n$ for
(5.1). Since the total curvature is intrinsic, any two of such Ricci surfaces
with different total curvatures cannot be isometric.

Now we are going to prove that two Ricci surfaces with the same total
curvature $-4\pi n$ but different pair $(a,b)$ in (5.1) could not be
isometric. Assuming that there is an isometry $\tau\mathrel{\mathop{\mathchar
58\relax}}M_{1}\rightarrow M_{2}$, where
$M_{1},M_{2}\simeq\mathbb{C}\backslash\\{0\\}$ are endowed with metrics
defined as in (5.1) with $(a_{1},b_{1})\neq(a_{2},b_{2})$. Then $\tau$ can be
lifted to a biholomorphic map $\tilde{\tau}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{C}\rightarrow\mathbb{C}$ through the universal covering map
$\textrm{exp}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{C}\rightarrow\mathbb{C}\backslash\\{0\\}$, hence
$\tilde{\tau}$ is actually an affine function. More precisely, there exists a
pair $(\alpha,\beta)\in\mathbb{C}^{2}$ such that

(5.16) $\left\\{\begin{aligned} &~{}g=e^{n(\alpha
u+\beta)}+a_{1}=e^{nu}+a_{2},\\\ &~{}\eta=b_{1}\alpha e^{-n(\alpha
u+\beta)}du=b_{2}e^{-nu}du.\\\ \end{aligned}\right.$

By comparing both sides of (5.16), we get $\alpha=1$, $a_{1}=a_{2}$,
$e^{n\beta}=1$ and $b_{1}=b_{2}$, contradiction. This completes the proof of
the theorem. ∎

The previous proof is inspired by Umehara and Yamada (see (Umehara and
Yamada,, 1993)). We give here another proof, which provides a sight of the
method that we will use for our further discussion.

###### Proof.

For the same reason, we suppose that the Weierstrass data have the forms as in
(5.2) and (5.3).

(1). If $\psi$ is holomorphic at $\infty$, then it should be a constant
function, thus $\eta=az^{-\mu-1}dz$ with $a\in\mathbb{C}$. This means that $f$
does not have poles on $\mathbb{C}\setminus\\{0\\}$. In fact, if $f$ has a
pole on $\mathbb{C}\setminus\\{0\\}$, then $\eta$ should have zero at this
point, thus $a=0$ and $\eta\equiv 0$. This leads to the result $Q\equiv 0$,
which is a contradiction to the condition of a catenoidal end. Therefore, $f$
is holomorphic on $\mathbb{C}$.

(i). If $f$ is also holomorphic at $\infty$, then $f$ must be a constant
function, hence $g=bz^{\mu}$ with $b\in\mathbb{C}$. In this case, we get a
catenoid or a catenoid cousin.

(ii). If $f$ has poles at $\infty$, it should be a polynomial

(5.17) $\displaystyle f(z)=\nu(z-a_{1})...(z-a_{p})$

with $\nu,a_{i}\in\mathbb{C}\setminus\\{0\\}$ and $p\in\mathbb{N}^{*}$. Let us
make a substitution $z=\frac{1}{w}$, then we get

$\left\\{\begin{aligned} &~{}g=\nu w^{-\mu-p}(1-a_{1}w)...(1-a_{p}w),\\\
&~{}\eta=-aw^{\mu-1}.\\\ \end{aligned}\right.$

The condition of catenoidal end at $\infty$ becomes

$\displaystyle\textrm{ord}\ Q=-p-2=-2,$

which results in $p=0$, contradiction.

(2). If $\psi$ has poles at $\infty$, it must have the form

(5.18) $\displaystyle\psi(z)=(z-b_{1})...(z-b_{q})$

with $b_{i}\in\mathbb{C}\setminus\\{0\\}$ and $q\in\mathbb{N}^{*}$. As a
consequence, $f$ has the expression

(5.19) $\displaystyle
f(z)=\frac{\xi(z-c_{1})...(z-c_{r})}{(z-b_{1})...(z-b_{q})},$

where $r\in\mathbb{N}$ and $\xi,c_{j}\in\mathbb{C}\setminus\\{0\\}$ satisfying
$b_{i}\neq c_{j}$ for all $(i,j)$. Replacing $z$ by $\frac{1}{w}$, we obtain

$\left\\{\begin{aligned} &~{}g=\xi
w^{-\mu-r+q}\frac{(1-c_{1}w)...(1-c_{r}w)}{(1-b_{1}w)...(1-b_{q}w)},\\\
&~{}\eta=-\lambda w^{\mu-2q-1}(1-b_{1}w)^{2}...(1-b_{q}w)^{2}.\\\
\end{aligned}\right.$

(i). If $-\mu-r+q\neq 0$, then the condition of catenoidal end at $\infty$
implies $-r-q-2=-2$, thus $r+q=0$, which is not possible.

(ii). The only case left is $\mu=q-r\in\mathbb{N}^{*}$. Applying
$\tilde{g}=\frac{g-\xi}{\bar{\xi}g+1}$, we get the following formula

(5.20)
$\displaystyle\tilde{g}=\xi\frac{(1-c_{1}w)...(1-c_{r}w)-(1-b_{1}w)...(1-b_{q}w)}{\xi\bar{\xi}(1-c_{1}w)...(1-c_{r}w)+(1-b_{1}w)...(1-b_{q}w)}.$

On the one side, it can be seen that $\textrm{ord}\ \tilde{g}\leq q$ because
$q>r$. On the other side, since $\textrm{ord}\ \tilde{\eta}=\textrm{ord}\
\eta=\mu-2q-1=-q-r-1$, we can compute

$\displaystyle\textrm{ord}\ d\tilde{g}=-2-\textrm{ord}\ \tilde{\eta}=q+r-1\geq
0,$

hence $\textrm{ord}\ \tilde{g}=q+r$. We deduce from this comparison that
$r=0$. Combining (5.20) with the fact that $\textrm{ord}\ \tilde{g}=q$, we may
observe that

$\left\\{\begin{aligned} &~{}I_{1}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{1\leq i\leq n}c_{i}=0,\\\ &~{}I_{2}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{1\leq i<j\leq n}c_{i}c_{j}=0,\\\ &~{}...\\\
&~{}I_{n-1}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{1\leq
i_{1}<i_{2}<...<i_{n-1}\leq n}c_{i_{1}}...c_{i_{n-1}}=0,\\\
&~{}I_{n}\mathrel{\mathop{\mathchar 58\relax}}=c_{1}c_{2}...c_{n}\neq
0.\end{aligned}\right.$

Consequently, the Weierstrass data take the same form as in (5.11), hence the
conclusion can be drawn by utilizing the same argument as in the first proof.
∎

From the proofs of this theorem, we may easily obtain the following
conclusion.

######  Corollary 5.3.

A complete non-positively curved Ricci surface
$M\simeq\mathbb{C}\backslash\\{0\\}$ whose total curvature is $-4\pi\mu$ with
$\mu\in\mathbb{R}^{*}_{+}\setminus\mathbb{N}$ must be isometric to a catenoid
cousin.

######  Remark 5.4.

Umehara and Yamada have proven in Umehara and Yamada, (1993) a similar result
for complete CMC-1 surfaces in the hyperbolic 3-space $\mathcal{H}^{3}$. Then
by the Lawson correspondence, these surfaces correspond to Ricci surfaces. It
is worthwhile to mention that theorem 5.2 gives actually more surfaces than
theirs. In fact, the main difference between our theorem and the result of
Umehara and Yamada is the catenoid case and that they have for the case (5.1)
a supplementary condition

(5.21) $\displaystyle n^{2}+4bn=m^{2}$

with $m\in\mathbb{N}^{*}$. Condition (5.21) implies that
$b=\frac{m^{2}-n^{2}}{4n}\in\mathbb{Q}$, but we just require $b$ to be a real
number. Then the previous argument says that we have indeed more surfaces.
Moreover, up to applying a homothety, our result implies that every complete
non-positively curved Ricci surface $M\simeq\mathbb{C}\backslash\\{0\\}$ with
two catenoidal ends admits an isometric CMC-1 immersion in $\mathcal{H}^{3}$.

######  Remark 5.5.

Troyanov has classified all the metrics of constant Gauss curvature 1 on
$\bar{\mathbb{C}}$ with two conical singularities at $0$ and $\infty$ (see
(Troyanov,, 1989)). In fact, he affirms that if $d\sigma^{2}$ is a such
metric, then there exists $\lambda\geq 0$ and $\beta>-1$, such that
$d\sigma^{2}$ has the expression

(5.22) $\displaystyle
d\sigma^{2}=(2+2\beta)^{2}\frac{|z|^{2\beta}|dz|^{2}}{(|1+\lambda
z^{\beta+1}|^{2}+|z|^{2\beta+2})^{2}},$

where $\beta$ is either an integer, or $\lambda=0$. In theorem 5.2, the
condition for catenoidal ends at $0$ and $\infty$ implies that we should have
a flat metric $|Q|=\nu\frac{|dz|^{2}}{|z|^{2}}$ with
$\nu\in\mathbb{R}^{*}_{+}$. Let us fix this flat metric, then each metric
$d\sigma^{2}$ in Troyanov’s theorem gives rise to a desired Ricci surface for
us (see (Moroianu and Moroianu,, 2015), Lemma 2.3). Conversely, for every
Ricci surface in theorem 5.2, since the induced flat metric $\sqrt{-K}g_{M}$
coincides with $|Q|$ which does not vanish on $\mathbb{C}\setminus\\{0\\}$,
its Gauss curvature $K$ has no zeroes on $\mathbb{C}\setminus\\{0\\}$. Hence
the metric $(-K)g_{M}$ is of constant Gauss curvature 1 with two conical
singularities at $0$ and $\infty$. It is a direct verification that
$(-K)g_{M}$ has the same expression as in (5.22). Therefore, our result and
Troyanov’s theorem are equivalent.

##  6\. $M\simeq\mathbb{C}\setminus\\{0,1\\}$ with three catenoidal ends

After giving a complete classification for non-positively curved Ricci surface
$M\simeq\mathbb{C}\setminus\\{0\\}$ with two catenoidal ends, we are naturally
interested in studying the case when the Ricci surface is topologically
$M\simeq\mathbb{C}\setminus\\{0,1\\}$ with three catenoidal ends at
$0,1,\infty$. However, since $\pi_{1}(M)\simeq\mathbb{Z}*\mathbb{Z}$ is not
abelian, we have to pay attention to the monodromy representation
$\rho\mathrel{\mathop{\mathchar 58\relax}}\pi_{1}(M)\rightarrow PSU(2)$, as
mentionned in the end of Section 4.

We will only consider the case when the induced metric $(-K)g_{M}$ is
reducible. Let us denote $\tau_{1},\tau_{2}$ the two generators of
$\pi_{1}(M)\simeq\mathbb{Z}*\mathbb{Z}$ which are represented by two closed
curves $\gamma_{1},\gamma_{2}$ surrounding $0$ and $1$, respectively. On the
one hand, since the image of $\rho$ is abelian, we may choose a proper $g$
such that

$\displaystyle g\circ{\tau_{j}}^{-1}=e^{2\pi i\theta_{j}}g,\quad j=1,2,$

where $\theta_{j}\in[0,1)$. Imitating the proof of lemma 4.4, one can verify
that the function $h(z)\mathrel{\mathop{\mathchar
58\relax}}=(z-1)^{-\theta_{2}}z^{-\theta_{1}}g$ is meromorphic on
$\bar{\mathbb{C}}$, thus a rational function. Therefore, $g$ should be written
as

$\displaystyle
g=(z-1)^{\theta_{2}+p}z^{\theta_{1}+q}\frac{\varphi(z)}{\psi(z)},$

where $p,q\in\mathbb{Z}$ and $\varphi,\psi$ are coprime polynomials without
zeroes at $0,1$. By interchanging the roles of $0,1$ and applying a rotation
if necessary, we may devide the Weierstrass data into two possible situations:
Case 1:

(6.1) $\left\\{\begin{aligned}
&~{}g=\lambda(z-1)^{\mu}z^{\nu}\frac{\varphi(z)}{\psi(z)},\\\
&~{}\eta=\kappa(z-1)^{-\mu-1}z^{-\nu-1}\psi^{2}(z)dz,\\\ \end{aligned}\right.$

where $\lambda,\kappa\in\mathbb{C}\setminus\\{0\\}$, $\mu>0$,
$\nu\in\mathbb{R}^{*}$ and $\varphi,\psi$ are coprime polynomials without
zeroes at $0,1$.
Case 2:

(6.2) $\left\\{\begin{aligned}
&~{}g=\lambda(z-1)^{\mu}\frac{\varphi(z)}{\psi(z)},\\\
&~{}\eta=\kappa(z-1)^{-\mu-1}z^{-\nu-1}\psi^{2}(z)dz,\\\ \end{aligned}\right.$

where $\lambda,\kappa\in\mathbb{C}\setminus\\{0\\}$, $\mu>0$,
$\nu\in\mathbb{N}^{*}$ and $\varphi,\psi$ are coprime polynomials without
zeroes at $0,1$.

In fact, if $\theta_{2}+p\neq 0$, then up to replacing $g$ by $\frac{1}{g}$,
we will obtain (6.1) or (6.2). In the case when $\theta_{2}+p=0$ but
$\theta_{1}+q\neq 0$, we may apply a Mobius transformation $\sigma(z)=1-z$ to
interchange the role of $0$ and $1$, then we are back to the first case. For
the last case, if both $\theta_{2}+p$ and $\theta_{1}+q$ are zero, then we may
use a rotation to get a new $g$ which has zeroes at $1$. Since such a rotation
does not change the orders of $dg$ and $\eta$ at $1$, we will find (6.1) or
(6.2) again.

On the other hand, we know from Section 4 that the Hopf differential $Q$ is
holomorphic on $M$ with three poles of order $2$ at $0,1$ and $\infty$, thus
it can be determined to be

(6.3) $\displaystyle
Q=\alpha\frac{(z-a_{1})(z-a_{2})}{z^{2}(z-1)^{2}}dz\otimes dz,$

where $a_{1},a_{2}\in\mathbb{C}\setminus\\{0,1\\}$ and
$\alpha\in\mathbb{C}\setminus\\{0\\}$. Combining (6.1), (6.2) with the
definition of Hopf differential $Q=dg\otimes\eta$, we can see that no matter
which case shall we take, $dg$ actually has a uniform expression

(6.4) $\displaystyle
dg=\frac{\alpha}{\kappa}(z-1)^{\mu-1}z^{\nu-1}\frac{(z-a_{1})(z-a_{2})}{\psi^{2}(z)}dz.$

Therefore, we obtain two differential equations with respect to the two cases:
For case 1, we have

(6.5)
$\displaystyle[(\mu+\nu)z-\nu]\varphi\psi+z(z-1)(\varphi^{\prime}\psi-\varphi\psi^{\prime})=\frac{\alpha}{\lambda\kappa}(z-a_{1})(z-a_{2}).$

For case 2, the equation is

(6.6)
$\displaystyle\mu\varphi\psi+(z-1)(\varphi^{\prime}\psi-\varphi\psi^{\prime})=\frac{\alpha}{\lambda\kappa}z^{\nu-1}(z-a_{1})(z-a_{2}).$

Any solution $\varphi,\psi$ of (6.5) or (6.6) will give rise to a desired
Ricci surface. However, these two equations are very complicated to solve
directly, so we need to do a more general discussion.

(1). Suppose that $a_{1}\neq a_{2}$. If $\psi$ has a zero $c$ of order higher
than $1$, then by analysing (6.5) and (6.6), we have
$(z-c)|(z-a_{1})(z-a_{2})$, hence $c=a_{1}$ or $c=a_{2}$ and the order of
$(z-c)$ is exactly $2$.

(i). If $\psi(z)=(z-a_{1})^{2}(z-a_{2})^{2}\widehat{\psi}$, then
$\widehat{\psi}$ should be a polynomial whose zeroes are all distinct. Let us
denote $\widehat{\psi}(z)=\prod_{j=1}^{n}(z-b_{j})$. The existence of $g$
requires that all the residues at poles of $dg$ (zeroes of $\psi$) to be zero,
whence we get the following conditions:

(6.7)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}-\frac{3}{b_{i}-a_{1}}-\frac{3}{b_{i}-a_{2}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\},$
$\displaystyle\frac{(\mu-1)(\mu-2)}{(a_{1}-1)^{2}}+\frac{2(\mu-1)(\nu-1)}{a_{1}(a_{1}-1)}+\frac{(\nu-1)(\nu-2)}{a_{1}^{2}}+\frac{12}{(a_{1}-a_{2})^{2}}$
(6.8) $\displaystyle-$
$\displaystyle\left(\frac{\mu-1}{a_{1}-1}+\frac{\nu-1}{a_{1}}\right)\left(\frac{6}{a_{1}-a_{2}}+\sum_{j=1}^{n}\frac{4}{a_{1}-b_{j}}\right)+\frac{6}{a_{1}-a_{2}}\left(\sum_{j=1}^{n}\frac{2}{a_{1}-b_{j}}\right)$
$\displaystyle+$
$\displaystyle\left(\sum_{j=1}^{n}\frac{2}{a_{1}-b_{j}}\right)^{2}+\sum_{j=1}^{n}\frac{2}{(a_{1}-b_{j})^{2}}=0,$

and

$\displaystyle\frac{(\mu-1)(\mu-2)}{(a_{2}-1)^{2}}+\frac{2(\mu-1)(\nu-1)}{a_{2}(a_{2}-1)}+\frac{(\nu-1)(\nu-2)}{a_{2}^{2}}+\frac{12}{(a_{2}-a_{1})^{2}}$
(6.9) $\displaystyle-$
$\displaystyle\left(\frac{\mu-1}{a_{2}-1}+\frac{\nu-1}{a_{2}}\right)\left(\frac{6}{a_{2}-a_{1}}+\sum_{j=1}^{n}\frac{4}{a_{2}-b_{j}}\right)+\frac{6}{a_{2}-a_{1}}\left(\sum_{j=1}^{n}\frac{2}{a_{2}-b_{j}}\right)$
$\displaystyle+$
$\displaystyle\left(\sum_{j=1}^{n}\frac{2}{a_{2}-b_{j}}\right)^{2}+\sum_{j=1}^{n}\frac{2}{(a_{2}-b_{j})^{2}}=0.$

(ii). Let us consider the case when $\psi(z)=(z-a_{1})^{2}\widehat{\psi}$,
where $\widehat{\psi}$ is the same as in case (i). A similar computation
yields that

(6.10)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}-\frac{3}{b_{i}-a_{1}}+\frac{1}{b_{i}-a_{2}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\},$

and

$\displaystyle\frac{(\mu-1)(\mu-2)}{(a_{1}-1)^{2}}+\frac{2(\mu-1)(\nu-1)}{a_{1}(a_{1}-1)}+\frac{(\nu-1)(\nu-2)}{a_{1}^{2}}+\frac{2(\nu-1)}{a_{1}(a_{1}-a_{2})}$
(6.11) $\displaystyle+$
$\displaystyle\frac{2(\mu-1)}{(a_{1}-1)(a_{1}-a_{2})}-\left(\sum_{j=1}^{n}\frac{4}{a_{1}-b_{j}}\right)\left(\frac{\mu-1}{a_{1}-1}+\frac{\nu-1}{a_{1}}+\frac{1}{a_{1}-a_{2}}\right)$
$\displaystyle+$
$\displaystyle\left(\sum_{j=1}^{n}\frac{2}{a_{1}-b_{j}}\right)^{2}+\sum_{j=1}^{n}\frac{2}{(a_{1}-b_{j})^{2}}=0,$

should be satisfied.

(iii). The last case is when all the zeroes of $\psi$ are distinct. We will
keep the notation as in the two previous cases, then we must have

(6.12)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}+\frac{1}{b_{i}-a_{1}}+\frac{1}{b_{i}-a_{2}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\}.$

(2). Now let us assume that $a_{1}=a_{2}$. In this case, if $\psi$ has a zero
with multiplicity, then it could be $(z-a_{1})^{2}$ or $(z-a_{1})^{3}$.

(i). For the case when $\psi$ does not have zeroes with multiplicity, we
easily get

(6.13)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}+\frac{2}{b_{i}-a_{1}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\}.$

(ii). If $\psi(z)=(z-a_{1})^{2}\widehat{\psi}$, then the conditions are

(6.14)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}-\frac{2}{b_{i}-a_{1}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\},$

and

(6.15)
$\displaystyle\frac{\mu-1}{a_{1}-1}+\frac{\nu-1}{a_{1}}-\sum_{j=1}^{n}\frac{2}{a_{1}-b_{j}}=0,\quad\forall
i\in\\{1,2,...,n\\}.$

(iii). The only case left is when $\psi(z)=(z-a_{1})^{3}\widehat{\psi}$.
Similarly, the conditions can be computed to be

(6.16)
$\displaystyle\frac{\mu-1}{b_{i}-1}+\frac{\nu-1}{b_{i}}-\frac{4}{b_{i}-a_{1}}-\sum_{j\neq
i}\frac{2}{b_{i}-b_{j}}=0,\quad\forall i\in\\{1,2,...,n\\},$

and

$\displaystyle
2\left(\frac{\mu-1}{a_{1}-1}+\frac{\nu-1}{a_{1}}\right)\left[11(\sum_{j=1}^{n}\frac{1}{a_{1}-b_{j}})^{2}+5\sum_{i\neq
j}\frac{1}{(a_{1}-b_{i})(a_{1}-b_{j})}\right]$ $\displaystyle+$ $\displaystyle
3\left[\frac{(\mu-1)(\mu-2)}{(a_{1}-1)^{2}}+\frac{2(\mu-1)(\nu-1)}{a_{1}(a_{1}-1)}+\frac{(\nu-1)(\nu-2)}{a_{1}^{2}}\right]\left(\sum_{j=1}^{n}\frac{2}{a_{1}-b_{j}}\right)$
(6.17) $\displaystyle+$
$\displaystyle\frac{(\mu-1)(\mu-2)(\mu-3)}{(a_{1}-1)^{3}}+\frac{3(\mu-1)(\nu-1)(\mu-2)}{a_{1}(a_{1}-1)^{2}}$
$\displaystyle+$
$\displaystyle\frac{3(\mu-1)(\nu-1)(\nu-2)}{a_{1}^{2}(a_{1}-1)}+\frac{(\nu-1)(\nu-2)(\nu-3)}{a_{1}^{3}}$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{n}\frac{32}{(a_{1}-b_{j})^{3}}-\sum_{i\neq
j}\frac{60}{(a_{1}-b_{i})^{2}(a_{1}-b_{j})}$ $\displaystyle-$
$\displaystyle\sum_{i\neq j\neq
k}\frac{16}{(a_{1}-b_{i})(a_{1}-b_{j})(a_{1}-b_{k})}=0.$

If a Ricci surface induces a reducible metric $(-K)g_{M}$ of constant Gauss
curvature 1, then it must satisfy equation (6.5) or (6.6) and one of the
previous cases; the converse is also true. Therefore, so as to find all the
desired Ricci surfaces, we may firstly solve algebraic equations (6.7)–(6) of
each case to completely determine the polynomial $\psi$. Then the problem is
reduced to searching solutions to first order equations (6.5) and (6.6) with a
known polynomial $\psi$. We summarize these arguments in the following
theorem:

######  Theorem 6.1.

Given $\mu>0$, $\nu\in\mathbb{R}^{*}$ and
$a_{1},a_{2}\in\mathbb{C}\setminus\\{0,1\\}$, a complete non-positively curved
Ricci surface $M\simeq\mathbb{C}\setminus\\{0,1\\}$ with a reducible metric
$(-K)g_{M}$ and three catenoidal ends exists if only if one of the following
cases is verified:

(i). If $a_{1}\neq a_{2}$, there exist $n\in\mathbb{N}$ and mutually distinct
$b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1},a_{2}\\}$ such that
(6.7)–(6) are satisfied and equation (6.5) or (6.6) admits a polynomial
solution $\varphi$ with
$\psi=(z-a_{1})^{2}(z-a_{2})^{2}\prod_{j=1}^{n}(z-b_{j})$;

(ii). If $a_{1}\neq a_{2}$, there exist $n\in\mathbb{N}$ and mutually distinct
$b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1},a_{2}\\}$ such that
(6.10)–(6) are satisfied and equation (6.5) or (6.6) admits a polynomial
solution $\varphi$ with $\psi=(z-a_{1})^{2}\prod_{j=1}^{n}(z-b_{j})$;

(iii). If $a_{1}\neq a_{2}$, there exist $n\in\mathbb{N}$ and mutually
distinct $b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1},a_{2}\\}$ such
that (6.12) are satisfied and equation (6.5) or (6.6) admits a polynomial
solution $\varphi$ with $\psi=\prod_{j=1}^{n}(z-b_{j})$;

(iv). If $a_{1}=a_{2}$, there exist $n\in\mathbb{N}$ and mutually distinct
$b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1}\\}$ such that (6.13) are
satisfied and equation (6.5) or (6.6) admits a polynomial solution $\varphi$
with $\psi=\prod_{j=1}^{n}(z-b_{j})$;

(v). If $a_{1}=a_{2}$, there exist $n\in\mathbb{N}$ and mutually distinct
$b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1}\\}$ such that (6.14)–(6.15)
are satisfied and equation (6.5) or (6.6) admits a polynomial solution
$\varphi$ with $\psi=(z-a_{1})^{2}\prod_{j=1}^{n}(z-b_{j})$;

(vi). If $a_{1}=a_{2}$, there exist $n\in\mathbb{N}$ and mutually distinct
$b_{1},...,b_{n}\in\mathbb{C}\setminus\\{0,1,a_{1}\\}$ such that (6.16)–(6)
are satisfied and equation (6.5) or (6.6) admits a polynomial solution
$\varphi$ with $\psi=(z-a_{1})^{3}\prod_{j=1}^{n}(z-b_{j})$.

######  Remark 6.2.

In the case when $M\simeq\mathbb{C}\setminus\\{0,1\\}$, we are not allowed to
use the method mentioned in remark 5.5 to study Ricci surfaces. This is
because the flat metric $|Q|$ vanishes at
$a_{1},a_{2}\in\mathbb{C}\setminus\\{0,1\\}$, which should coincide with
zeroes of the Gauss curvature $K$. It means that the hypothesis $V>0$ of Lemma
2.3 in (Moroianu and Moroianu,, 2015) fails to be satisfied. Therefore, we are
obliged to take advantage of the method as above.

Now we plan to use theorem 6.1 to do some classification for Ricci surfaces
with a reducible metric $(-K)g_{M}$ and with finite total curvature $-4\pi$.
Denote $\textrm{deg}\ \psi=N$ and $\textrm{deg}\ \varphi=m$, then by using
Gauss-Bonnet formula with conical singularities (5.15) again, we are able to
compute the total curvature. The expression of total curvature should be
divided into several cases:

(1). Suppose that the Weierstrass data take the form as in (6.1) with $\nu>0$.
The induced metric $(-K)g_{M}$ can be determined to be

$\displaystyle(-K)g_{M}=\frac{4|\lambda\alpha|^{2}}{|\kappa|^{2}}\cdot\frac{|z-1|^{2(\mu-1)}|z|^{2(\nu-1)}|z-a_{1}|^{2}|z-a_{2}|^{2}}{(|\psi|^{2}+|\lambda|^{2}|z-1|^{2\mu}|z|^{2\nu}|\varphi|^{2})^{2}}|dz|^{2}.$

(a). If $N>\mu+\nu+m$, then this metric has conical sinularities of order
$\nu-1,\mu-1,2N-2-\mu-\nu,1,1$ at $0,1,\infty,a_{1},a_{2}$, respectively.
Hence the total curvature is $-4\pi N$. The hypothesis that the total
curvature should be $-4\pi$ implies $N=1$, thus $m=0$ and $0<\mu+\nu<1$. In
this case, we have $\varphi=1$ and $\psi(z)=z-b$ for some
$b\in\mathbb{C}\setminus\\{0,1\\}$. Comparing two sides of equation (6.5), we
can see that if

(6.18)
$\displaystyle\nu(\nu-1)=(\mu+\nu-1)[\nu(a_{1}+a_{2})-(\mu+\nu)a_{1}a_{2}]$

is verified, then $b=\frac{(\mu+\nu-1)a_{1}a_{2}}{\nu}$ is uniquely
determined. Moreover, theorem 6.1 tells us that $b$ should satisfy condition
(6.12) or (6.13).

(b). If $N=\mu+\nu+m$ and $N+m\geq 2$, then the total curvature is also $-4\pi
N$. Hence we must have $N=1$ and $m=0$, which is not possible.

(c). If $N<\mu+\nu+m$, then the total curvature is $-4\pi(\mu+\nu+m)$. The
condition $\mu+\nu+m=1$ tells us that $m=0$ and $N=0$. In this case,
$\psi=\varphi=1$ and that condition (6.12) or (6.13) is obviously satisfied.
However, equation (6.5) will not hold because the degrees on the two sides are
not coincident. Hence this case cannot appear.

(2). Now assume that the Weierstrass data also take the form as in (6.1) but
with $\nu<0$. In this case, we have

$\displaystyle(-K)g_{M}=\frac{4|\lambda\alpha|^{2}}{|\kappa|^{2}}\cdot\frac{|z-1|^{2(\mu-1)}|z|^{-2(\nu+1)}|z-a_{1}|^{2}|z-a_{2}|^{2}}{(|z|^{-2\nu}|\psi|^{2}+|\lambda|^{2}|z-1|^{2\mu}|\varphi|^{2})^{2}}|dz|^{2}.$

(d). If $N>\mu+\nu+m$, then the total curvature is $-4\pi(N-\nu)$, thus we get
$N=0$ and $\nu=-1$. Moreover, $1>\mu+m$ implies $m=0$. The same argument as in
(c) shows that this is impossible.

(e). If $N<\mu+\nu+m$, then the total curvature is $-4\pi(\mu+m)$. Again, we
will obtain $m=0$ and $N=0$, not possible.

(f). If $N=\mu+\nu+m$ and $N+m\geq 2$, then the total curvature is also
$-4\pi(N-v)$. This leads to $N=m=0$, impossible.

(3). Suppose now that the Weierstrass data have the expression as in (6.2),
where $\nu\in\mathbb{N}^{*}$. A similar computation shows

$\displaystyle(-K)g_{M}=\frac{4|\lambda\alpha|^{2}}{|\kappa|^{2}}\cdot\frac{|z-1|^{2(\mu-1)}|z|^{2(\nu-1)}|z-a_{1}|^{2}|z-a_{2}|^{2}}{(|\psi|^{2}+|\lambda|^{2}|z-1|^{2\mu}|\varphi|^{2})^{2}}|dz|^{2}.$

(g). If $N>\mu+m$, the total curvature is $-4\pi N$, thus $N=1$ and $m=0$. In
this case, $\varphi=1$ and $\psi(z)=z-b$ for some $b$ satisfying (6.12) or
(6.13). However, the degree of the left-hand side of (6.6) is $1$ while the
degree of the right-hand side is at least $2$, contradiction.

(h). If $N<\mu+m$, the total curvature is $-4\pi(\mu+m)$. This implies
$m=N=0$. A similar argument as in (c) shows that equation (6.6) will never
hold, thus it is impossible.

(i). If $N=\mu+m$ and $N+m\geq\nu+2$, the total curvature is $-4\pi N$. In
this case, we get $N=1$ and $m=0$, but $N+m=1$ contradicts to the condition
$N+m\geq\nu+2$.

These arguments can be concluded as the following theorem:

######  Theorem 6.3.

Assuming that $M\simeq\mathbb{C}\setminus\\{0,1\\}$ is a complete Ricci
surface with three catenoidal ends and a reducible induced metric $(-K)g_{M}$,
if it has finite total curvature $-4\pi$, then it must be isometric to the
surface determined by the Weierstrass data

(6.19) $\left\\{\begin{aligned}
&~{}g=\lambda(z-1)^{\mu}z^{\nu}\frac{1}{z-\frac{(\mu+\nu-1)a_{1}a_{2}}{\nu}},\\\
&~{}\eta=\kappa(z-1)^{-\mu-1}z^{-\nu-1}\left(z-\frac{(\mu+\nu-1)a_{1}a_{2}}{\nu}\right)^{2}dz,\\\
\end{aligned}\right.$

where $a_{1},a_{2}\in\mathbb{C}\setminus\\{0,1\\}$,
$\lambda,\kappa\in\mathbb{C}\setminus\\{0\\}$, $\mu,\nu\in\mathbb{R}^{*}_{+}$
satisfying $0<\mu+\nu<1$, (6.18) and (6.12) (or (6.13)).

Moreover, we get instantly the following result from the previous discussion.

######  Theorem 6.4.

There is no complete Ricci surface $M\simeq\mathbb{C}\setminus\\{0,1\\}$ with
three catenoidal ends which induces a reducible metric $(-K)g_{M}$ and has
finite total curvature $-4\pi l$ for $0<l<1$.

With the help of theorem 6.3, we may construct an example.

######  Example 6.5.

Taking $\mu=\frac{1}{4}$, $\nu=\frac{1}{2}$, then condition (6.18) becomes

$\displaystyle 4=2(a_{1}+a_{2})-3a_{1}a_{2}.$

By a direct computation, we can verify that (6.12) or (6.13) is true for
$b=-\frac{1}{2}a_{1}a_{2}$. Therefore, we may take $a_{1}=\frac{6}{5}$ and
$a_{2}=-1$, then (6.19) will give us a desired example of Ricci surface.

For Ricci surfaces $M\simeq\mathbb{C}\setminus\\{0,1\\}$ with total curvature
$-4\pi l$ for $l>1$, things will become much more complicated. However, it is
always beneficial to mention the trinoid constructed by Jorge and Meeks in
(Jorge and Meeks III,, 1983), which is a typical example of such Ricci
surfaces whose induced metric $(-K)g_{M}$ is reducible.

######  Example 6.6.

(Trinoid) The classical Weierstrass data of a trinoid are

$\displaystyle g=z^{2},\quad\eta=\frac{dz}{(z^{3}-1)^{2}},$

defined on $\bar{\mathbb{C}}\setminus\\{1,\xi,\xi^{2}\\}$, where
$\xi=e^{\frac{2\pi i}{3}}$. After a Mobius transformation and a rotation, we
may find a new pair of Weierstrass data defined on
$\mathbb{C}\setminus\\{0,1\\}$ as

(6.20) $\left\\{\begin{aligned}
&~{}g=\frac{\xi-1}{\xi+1}\cdot\frac{(w-1)(w+1)}{(w-2-\sqrt{3})(w-2+\sqrt{3})},\\\
&~{}\eta=\frac{-1}{9(2\xi+1)}\cdot\frac{(w-2-\sqrt{3})^{2}(w-2+\sqrt{3})^{2}}{w^{2}(w-1)^{2}}dw.\\\
\end{aligned}\right.$

Then a little computation shows $a_{1}=\frac{1+\sqrt{3}i}{2}$ and
$a_{2}=\frac{1-\sqrt{3}i}{2}$. It can be checked that conditions (6.12) and
equation (6.6) are satisfied. Additionally, by using Gauss-Bonnet formula
(5.15) again, we know that its total curvature is $-8\pi$.

##  7\. Higher genus Ricci surfaces

As we have seen in Section 6, it is very difficult to give a total
classification of Ricci surfaces with catenoidal ends in general. However, the
existence of Ricci surfaces
$M\simeq\bar{\mathbb{C}}\setminus\\{p_{1},p_{2},...,p_{n}\\}$ with $n$
catenoidal ends ($n\geq 2$) is well-known, which is given by the n-noid (see
Weber, (2005)).

######  Example 7.1.

(n-noid) Suppose $M\simeq\bar{\mathbb{C}}\setminus\\{1,\xi,...,\xi^{n-1}\\}$
with $\xi=e^{\frac{2\pi i}{n}}$, then the Weierstrass data of the n-noid are
given by

$\displaystyle g=z^{n-1},\quad\eta=\frac{dz}{(z^{n}-1)^{2}}.$

Now we are interested in discussing non-positively curved Ricci surfaces with
positive genus and finitely many catenoidal ends. Our goal of this section is
to prove that there exist Ricci surfaces $M\simeq
S_{k}\setminus\\{p_{1},p_{2},...,p_{n}\\}$ with $n$ catenoidal ends for
$k,n>0$, where $S_{k}$ is a compact orientable surface of genus $k$. To
achieve this, we will use the method introduced by Andrei Moroianu and Sergiu
Moroianu as mentioned in remark 5.5. The first tool that we need is the
following theorem proved by Gabriele Mondello and Dmitri Panov (see Mondello
and Panov, (2019), Theorem A).

######  Theorem 7.2.

Let $\chi\leq 0$ be an even number and $v_{1},...,v_{n}\in\mathbb{R}^{*}_{+}$
be such that

(7.1) $\displaystyle\chi+\sum_{j=1}^{n}(v_{j}-1)>0,$

then there exists a compact Riemann surface $S_{k}$ of genus
$k=\frac{2-\chi}{2}\geq 1$, a set of distinct points
$\\{p_{1},p_{2},...,p_{n}\\}\subset S_{k}$ and a metric $g_{1}$ of constant
Gauss curvature 1 on $S_{k}$ such that $g_{1}$ has conical singularities of
order $v_{j}-1$ at $p_{j}$.

In order to get a Ricci metric on $M\simeq
S_{k}\setminus\\{p_{1},p_{2},...,p_{n}\\}$, we have to construct a proper flat
metric. This is possible due to a result of Andrei Moroianu and Sergiu
Moroianu (see (Moroianu and Moroianu,, 2015), Lemma 6.1).

######  Lemma 7.3.

Given a compact Riemann surface $S_{k}$ of genus $k$, let
$p_{1},p_{2},...,p_{n}\in S_{k}$ and $\beta\mathrel{\mathop{\mathchar
58\relax}}\\{p_{1},p_{2},...,p_{n}\\}\rightarrow\mathbb{R}$ be a function
which satisfies

(7.2) $\displaystyle\sum_{j=1}^{n}(\beta(p_{j})-1)=-\chi(S_{k}),$

then there exists a flat metric $g_{0}$ on
$S_{k}\setminus\\{p_{1},p_{2},...,p_{n}\\}$ compatible with the complex
structure of $S_{k}$ such that near each $p_{i}$, it has the form

(7.3) $\displaystyle g_{0}=e^{2u}|z|^{2\beta(p_{i})-2}|dz|^{2}$

with some smooth function $u\in C^{\infty}(S_{k},\mathbb{R})$.

Thanks to these two results, we are able to prove the existence of Ricci
surfaces with arbitrary genus and arbitrary number of catenoidal ends.

######  Theorem 7.4.

For $k,n>0$, there exists a non-positively curved orientable Ricci surface $M$
of genus $k$ with $n$ cetenoidal ends.

###### Proof.

Firstly, let us take $\chi=2-2k\leq 0$, $v_{1},...,v_{n}\in\mathbb{R}^{*}_{+}$
and $v_{n+1}=2(2k-2+n)+1>0$, then it is easy to verify that

$\displaystyle\chi+\sum_{j=1}^{n+1}(v_{j}-1)=\sum_{j=1}^{n}v_{j}+2k+n-2>0.$

Hence by theorem 7.2, we will have a compact Riemann surface $S_{k}$ of genus
$k$, a set of points $\\{p_{1},p_{2},...,p_{n+1}\\}\subset S_{k}$ and a metric
$g_{1}$ of constant Gauss curvature 1 on $S_{k}$ with conical singularities of
order $v_{j}-1$ at $p_{j}$.

Secondly, we define a function $\beta\mathrel{\mathop{\mathchar
58\relax}}\\{p_{1},p_{2},...,p_{n+1}\\}\rightarrow\mathbb{R}$ as
$\beta(p_{i})=0$ for $i=1,2,...,n$ and $\beta(p_{n+1})=2k-1+n$. This function
$\beta$ satisfies the equality (7.2), thus from lemma 7.3, there is a flat
metric $g_{0}$ on $S_{k}\setminus\\{p_{1},p_{2},...,p_{n+1}\\}$ which is
conformal to $g_{1}$ and takes the form as in (7.3) near every punctured
point.

Both of $g_{1}$ and $g_{0}$ do not vanish on
$S_{k}\setminus\\{p_{1},p_{2},...,p_{n+1}\\}$. Since $g_{0}$ is conformal to
$g_{1}$, there is a positive function $V$ defined on
$S_{k}\setminus\\{p_{1},p_{2},...,p_{n+1}\\}$ such that $g_{1}=Vg_{0}$. It
follows that $g\mathrel{\mathop{\mathchar 58\relax}}=V^{-1}g_{0}$ is a Ricci
metric on $S_{k}\setminus\\{p_{1},p_{2},...,p_{n+1}\\}$ (see (Moroianu and
Moroianu,, 2015), Lemma 2.3). Near the point $p_{n+1}$, it is known that the
metric $g_{1}$ can be written as

(7.4) $\displaystyle
g_{1}=\frac{4v_{n+1}^{2}|z|^{4(2k-2+n)}|dz|^{2}}{(1+|z|^{4(2k-2+n)+2})^{2}}$

for a suitable complex coordinate $z$. From lemma 7.3, the flat metric $g_{0}$
has also a local expression around $p_{n+1}$ which is

(7.5) $\displaystyle g_{0}=e^{2u}|z|^{2(2k-2+n)}|dz|^{2}$

for some smooth function $u$. A direct computation shows that the Ricci metric
$g$ is of the form

(7.6) $\displaystyle
g=\frac{1}{4}v_{n+1}^{-2}e^{4u}(1+|z|^{4(2k-2+n)+2})^{2}|dz|^{2}$

near $p_{n+1}$. Hence $g$ is actually well-defined and smooth at $p_{n+1}$,
thus a Ricci metric on $S_{k}\setminus\\{p_{1},p_{2},...,p_{n}\\}$. In
addition, near each $p_{i}\in\\{p_{1},p_{2},...,p_{n}\\}$, $g_{0}$ has the
form

$\displaystyle g_{0}=e^{2u}|z|^{-2}|dz|^{2}.$

This means $p_{1},p_{2},...,p_{n}$ are catenoidal ends of this Ricci surface
(see the discussion after definition 4.3). Consequently, we have constructed
an orientable Ricci surface of genus $k$ with exactly $n$ catenoidal ends. ∎

######  Example 7.5.

Wayne Rossman and Katsunori Sato have constructed a genus 1 catenoid cousin
which is a CMC-1 immersion into the hyperbolic 3-space $\mathcal{H}^{3}$ (see
Rossman and Sato, (1998)). This is an example of theorem 7.4 in the case $k=1$
and $n=2$.

## References

  * Bryant, [1987] Bryant, R. (1987).  Surfaces of mean curvature one in hyperbolic space.  Astérisque, 154(155):321–347.
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