#  Classification of non-CSC extremal Kähler metrics on K-surfaces
$S^{2}_{\\{\alpha\\}}$ and $S^{2}_{\\{\alpha,\beta\\}}$

Yingjie Meng, Zhiqiang Wei

###### Abstract

We commonly refer to an extremal Kähler metric with finitely many
singularities on a compact Riemann surface as an HCMU (Hessian of the
Curvature of the Metric is Umbilical) metric. In this study, we specifically
classify non-CSC HCMU metrics on the K-surfaces $S^{2}_{\\{\alpha\\}}$ and
$S^{2}_{\\{\alpha,\beta\\}}$.

Key words extremal Kähler metric, Conical singularity, Cusp singularity.

2020 MR Subject Classification: 58E11, 53C55.

##  1 Introduction

As is well known, within a given Kähler class of a compact Kähler manifold
$\mathcal{M}$, an extremal Kähler metric, introduced by E. Calabi in [1], is
the critical point of the following Calabi energy functional

$\mathcal{C}(g)=\int_{\mathcal{M}}R^{2}dg,$

where $R$ denotes the scalar curvature of the metric $g$ in the Kähler class.
The objective is to determine the “best” metric within a fixed Kähler class.
The Euler-Lagrange equations for the functional $\mathcal{C}(g)$ are given by
$R_{,\alpha\beta}=0$ for all indices $\alpha,\beta$, with $R_{,\alpha\beta}$
being $(2,0)$ part of the second covariant derivative of $R$. When
$\mathcal{M}$ is a compact Riemann surface, Calabi [1] proved that an extremal
Kähler metric is of constant scalar curvature (CSC) metric. This coincides
with the classical uniformization theorem, which asserts that every Riemann
surface admits a CSC metric.

A natural question then arises: do extremal Kähler metrics with singularities
on a compact Riemann surface still possess a CSC property? This inquiry
represents an attempt to generalize the classical uniformization theorem to a
K-surface. The existence or nonexistence of CSC metrics on surfaces with
conical singularities has been extensively studied in works such as [7, 11,
12, 15, 19, 17, 20] and further references cited therein. In [4], X.X. Chen
made a significant breakthrough by presenting an example of a non-CSC extremal
Kähler metric with singularities. He also classified all extremal Kähler
metrics on compact Riemann surfaces possessing finite cusp singularities and
having finite area and energy.

######  Theorem 1.1 ([4]).

Let $\mathcal{M}$ be a compact Riemann surface, $g$ an extremal Kähler metric
with finite energy and area on
$\mathcal{M}\setminus\\{p_{j}\\}_{j=1,\ldots,n}$, and $K$ the Gaussian
curvature of $g$. Suppose that all singularities $p_{j}$ are weak cusps. Then
the following classification holds.

(i) If genus$(\mathcal{M})\geq 1$, then $K\equiv Const$.

(ii) If $\mathcal{M}=S^{2}$ and $n\geq 3$, then $K\equiv Const$.

(iii) If $\mathcal{M}=S^{2}$ and $n=2$, then there is no extremal Kähler
metric.

(iv) If $\mathcal{M}=S^{2}$ and $n=1$, then $g$ is a rotationally symmetric
metric that is determined uniquely by the total area.

In particular, $g$ is a metric with cusps.

In [21], G.F. Wang and X.H. Zhu discovered that every singular point of an
extremal metric with finite energy and area on a singular surface is either a
weak cusp or a conical singularity. They then generalized Theorem 1.1 as
follows.

######  Theorem 1.2 ([21]).

Let $\mathcal{M}$ be a compact Riemann surface, $g$ an extremal Kähler metric
with finite energy and area on
$\mathcal{M}\setminus\\{p_{j}\\}_{j=1,\ldots,n}$, and $K$ the Gaussian
curvature of $g$. Then $g$ is a conical metric with singular angle
$2\pi\alpha_{j}~{}(j=1,\ldots,n)$(which may include some weak cusps).
Furthermore, if all singular angles satisfy $2\pi\alpha_{j}<\pi$, then the
following classification holds.

(i) If genus$(\mathcal{M})\geq 1$, then $K\equiv Const$.

(ii) If $\mathcal{M}=S^{2}$ and $n\geq 3$, then $K\equiv Const$.

(iii) If $\mathcal{M}=S^{2}$ and $n=2$, there are two cases.

(a) If both singular points are cusp, then there is no extremal Kähler metric.

(b) If one of the singular points is not a cusp, then $g$ is a rotationally
symmetric extremal Kähler metric that is determined uniquely by the total area
and two angles $2\pi\alpha_{j}$.

(iv) If $\mathcal{M}=S^{2}$ and $n=1$, then $g$ is a rotationally symmetric
metric that is determined uniquely by the total area and angle $2\pi\alpha$.

Now, we commonly refer to an extremal Kähler metric with finitely many
singularities on a compact Riemann surface as an HCMU (Hessian of the
Curvature of the Metric is Umbilical) metric. In [5], X.X. Chen presented a
theorem, famously known as the obstruction theorem for non-CSC HCMU metrics
with conical singularities.

######  Theorem 1.3 ([5]).

Let $g$ be a non-CSC HCMU metric in a K-surface
$\mathcal{M}_{\\{\alpha_{1},\ldots,\alpha_{N}\\}}$. Then the Euler character
of the underlying surface should be determined by

$\chi(\mathcal{M})=\sum_{j=1}^{J}(1-\alpha_{j})+(N-J)+S$

where $2\pi\alpha_{1},\ldots,2\pi\alpha_{J}$ are all the singular angles
corresponding to the saddle points of the Gaussian curvature $K$ of $g$, and
$S$ is the number of smooth critical points of $K$.

According to Theorem 1.3, if the curvature $K$ of a non-CSC HCMU metric has no
saddle points, the underlying surface must be a sphere, and $K$ possesses
exactly two extremal points. Such an HCMU metric exhibits rotational
symmetric, which is called a football. Furthermore, Chen’s classification in
[5] details the properties of football metrics.

######  Theorem 1.4 ([5]).

If none of the prescribed angles in a K-surface is an integer multiple of
$2\pi$, then this K-surface supports a non-CSC HCMU metric if and only if it
is a football with two different angles at the two poles. Furthermore, any
non-CSC HCMU metric in these footballs must be rotationally symmetric and
uniquely determined by the area.

In [16], C.S. Lin and X.H. Zhu introduced a class of non-CSC HCMU metrics on
$S^{2}$ with finitely many conical singularities of angles $2\pi\cdot
integers$. These special non-CSC HCMU metrics are called exceptional when all
singularities are saddle points of the Gaussian curvature $K$. A minimal
exceptional HCMU metric is one where $K$ has a single minimum. They provided
an explicit formula for such metrics, showing they are determined by 3
parameters. In [8], Q. Chen and Y.Y. Wu derived an explicit formula for non-
CSC HCMU metrics on $S^{2}$ and $T^{2}$, generalizing the results from [16].
In [6], Q. Chen, X.X. Chen and Y.Y. Wu proved that non-CSC HCMU metrics are
locally isometric to a football, and reduced the existence question to a
combinatorial problem. For $\mathcal{M}=S^{2}$, they showed that Chen’s
obstruction theorem is sufficient for the existence of non-CSC HCMU metrics.
In [9, 10], the authors further reduced the existence problem to a meromorphic
1-form on the underlying Riemann surface. However, the existence of such a
form remains challenging. On $S^{2}$, this problem can be translated into an
algebraic one, which led to a class of non-CSC HCMU metrics with conical
singularities whose existence is independent of singularity positions, as
demonstrated in [23]. In [24], Y.Y. Wu and the second author classified
normalized non-CSC HCMU metrics on the K-surface $S^{2}_{\\{2,2,2\\}}$.
Normalization in this context means that the Gaussian curvature $K$ of the
metric is constrained such that its maximum is equal to $1$.

As far as we know, the classification of non-CSC HCMU metrics on $S^{2}$ with
conical singularities —even those with a small number —remains unexplored.
Therefore, this paper focuses on the classification problem for non-CSC HCMU
metrics with finite area and energy on K-surfaces $S^{2}_{\\{\alpha\\}}$ and
$S^{2}_{\\{\alpha,\beta\\}}$. Our results are as follows.

######  Theorem 1.5.

Suppose $g$ is a non-CSC HCMU metric on $S^{2}_{\\{\alpha\\}}$ and $K$ is the
Gaussian curvature of $g$, then the following classification holds.
(1) If the singularity is an extremal points of $K$, then $g$ is a
rotationally symmetric metric that is uniquely determined by the total area
and the angle $2\pi\alpha$. Explicitly, if $0\leq\alpha<1$, the singularity is
the minimum point of $K$; if $\alpha>1$, the singularity is the maximum point
of $K$.
(2) If the singularity is the saddle point of $K$, then
$2\leq\alpha\in\mathbb{Z}$ and $K$ has $I_{1}\geq 1$ maximum points and
$I_{2}\geq 1$ minimum points, where $I_{1}$ and $I_{2}$ satisfy
$I_{1}>I_{2},I_{1}+I_{2}=\alpha+1$ and one of the following conditions holds.

(2-1) $I_{2}=1$.

(2-2) $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$.

######  Theorem 1.6.

Suppose $g$ is a non-CSC HCMU metric on $S^{2}_{\\{\alpha,\beta\\}}$ and $K$
is the Gaussian curvature of $g$, then the following classification holds.
(1) If two singularities are extremal points of $K$, then $g$ is a
rotationally symmetric metric that is uniquely determined by the total area
and the angles $2\pi\alpha,2\pi\beta$, i.e., g is a football.
(2) If one singularity is the saddle point of $K$ and the other is an extremal
point of $K$, without loss of generality, suppose the singularity of conical
angle $2\pi\alpha~{}(2\leq\alpha\in\mathbb{Z})$ is the saddle point of $K$,
then the following statements hold.

(A) If $2\leq\beta\in\mathbb{Z}$, there are two cases.

(A-1) If the singularity of singular angle $2\pi\beta$ is a maximum point of
$K$, then $K$ has $I_{1}+1\geq 1$ maximum points and $I_{2}\geq 1$ minimum
points, where $I_{1}$ and $I_{2}$ satisfy $I_{1}+I_{2}=\alpha$ and one of the
following conditions holds.

(a-1-1) $I_{2}=1$.

(a-1-2) $I_{2}\geq 2,I_{2}\mid(I_{1}+\beta)$ and $I_{1}+\beta>\alpha-1$.

(a-1-3) $I_{2}\geq 2$ and $I_{2}\nmid(I_{1}+\beta)$.

(A-2) If the singularity of singular angle $2\pi\beta$ is a minimum point of
$K$, then $K$ has $I_{1}\geq 1$ maximum points and $I_{2}+1\geq 1$ minimum
points, where $I_{1}$ and $I_{2}$ satisfy $I_{1}+I_{2}=\alpha$ and one of the
following conditions holds.

(a-2-1) $I_{2}=0$.

(a-2-2) $I_{2}\geq 1,(I_{2}+\beta)\mid I_{1}$ and
$I_{1}(I_{2}+\beta)\geq(\alpha-1){\rm GCD}(I_{2}+\beta,I_{1})$.

(B) If $\beta\notin\mathbb{Z}$, there are two cases.

(B-1) If the singularity of singular angle $2\pi\beta$ is a maximum point of
$K$, then $K$ has $I_{1}+1\geq 1$ maximum points and $I_{2}\geq 1$ minimum
points, where $I_{1}$ and $I_{2}$ satisfy $I_{1}+I_{2}=\alpha$ and
$I_{1}+\beta>I_{2}$.

(B-2) If the singularity of singular angle $2\pi\beta$ is a minimum point of
$K$, then $K$ has $I_{1}\geq 1$ maximum points and $I_{2}+1\geq 1$ minimum
points, where $I_{1}$ and $I_{2}$ satisfy $I_{1}+I_{2}=\alpha$ and
$I_{1}>I_{2}+\beta$.
(3) If both singularities are saddle points of $K$, then
$2\leq\alpha,\beta\in\mathbb{Z}$ and $K$ has $I_{1}\geq 1$ maximum points and
$I_{2}\geq 1$ minimum points, where $I_{1}$ and $I_{2}$ satisfy
$I_{1}>I_{2},I_{1}+I_{2}=\alpha+\beta$ and one of the following conditions
holds.

(3-1) $I_{2}=1$.

(3-2) $I_{2}\geq 2,I_{2}\mid I_{1}$ and $I_{1}\geq max\\{\alpha,\beta\\}$.

(3-3) $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$.

##  2 Preliminaries

###  2.1 HCMU metrics

######  Definition 2.1 ([21]).

(1) Let $g=e^{2\psi}|dz|^{2}$ be an extremal metric on punctured disk
$\mathbb{D}\setminus\\{0\\}$, where $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$.
The singular point $z=0$ is called a weak conical point with singular angle
$2\pi\alpha$ if and only if $\psi$ satisfies

$\lim_{r\rightarrow
0}\frac{1}{2\pi}\int_{0}^{2\pi}(r\frac{\partial\psi(r,\theta)}{\partial
r}+1-\alpha)d\theta=0.$ (1)

If $\alpha=0$ in (1), then the singular point $z=0$ is called a weak cusp.

(2) If $\psi$ can be locally expressed as

$\psi(z)=(\alpha-1)\ln|z|+\rho(z)$

with $\rho(z)$ a smooth function on $\mathbb{D}$ and $\alpha>0$, then the
singular point $z=0$ is called a conical point with singular angle
$2\pi\alpha$.

(3) If $\psi$ can be locally expressed as

$\psi(z)=-\ln|z|+\ln\rho(z)$

with $\rho(z)$ a smooth positive function on $\mathbb{D}$, then the singular
point $z=0$ is called a cusp point.

######  Definition 2.2 ([5]).

Let $\mathcal{M}$ be a compact Riemann surface and $p_{1},\cdots,p_{N}$ be $N$
points on $\mathcal{M}$. Denote
$\mathcal{M}\backslash\\{p_{1},\ldots,p_{N}\\}$ by $\mathcal{M}^{*}$. Let $g$
be a conformal metric on $\mathcal{M}^{*}$. If $g$ satisfies

$K_{,zz}=0,$ (2)

where $K$ is the Gaussian curvature of $g$, then we call $g$ an HCMU (Hessian
of the Curvature of the Metric is Umbilical) metric on $\mathcal{M}$.

In this paper, we restrict our attention to non-CSC HCMU metrics with finite
area and finite Calabi energy, that is,

$\int_{\mathcal{M}^{*}}dg<+\infty,~{}~{}\int_{\mathcal{M}^{*}}K^{2}dg<+\infty.$
(3)

From references [3, 5, 21], we know that each singularity of a non-CSC HCMU
metric is conical or cusp if it has finite area and finite Calabi energy.
Here, we summarize some key results of non-CSC HCMU metrics, which will be
used in this paper. First the equation (2) is equivalent to

$\nabla K=\sqrt{-1}e^{-2\varphi}K_{\bar{z}}\frac{\partial}{\partial z}$ (4)

which is a holomorphic vector field on $\mathcal{M}^{*}$. Independently,
authors in [5] and [16] demonstrated that the curvature $K$ can be extended
continuously to $\mathcal{M}$ and possesses a finite set of smooth extremal
points on $\mathcal{M}^{*}$. Further, [6] and [10] established that each
smooth extremum of $K$ is either a global maximum (denoted $K_{1}$) or a
global minimum (denoted $K_{2}$). Importantly, when all the singularities of
$g$ are conical singularities,

$K_{1}>0,~{}K_{1}>K_{2}>-(K_{1}+K_{2});$

when there exist cusps in the singularities,

$K_{1}>0,~{}K_{2}=-\frac{1}{2}K_{1}.$

In [16], C.S. Lin and X.H. Zhu demonstrated that $\nabla K$ is actually a
meromorphic vector field on $\mathcal{M}$. In [9, 10], the authors introduced
the dual 1-form of $\nabla K$ by $\omega(\nabla K)=\frac{\sqrt{-1}}{4}$. They
call $\omega$ the character 1-form of the metric, which plays a significant
role in studying non-CSC HCMU metrics. Now, denote
$\mathcal{M}^{*}\setminus\\{\text{smooth extremal points of}~{}K\\}$ by
$\mathcal{M}^{\prime}$. Then on $\mathcal{M}^{\prime}$

$\begin{cases}\cfrac{dK}{-\frac{1}{3}(K-K_{1})(K-K_{2})(K+K_{1}+K_{2})}=\omega+\bar{\omega},\\\
g=-\frac{4}{3}(K-K_{1})(K-K_{2})(K+K_{1}+K_{2})\omega\bar{\omega}.\\\
\end{cases}$ (5)

From (5), some properties of $\omega$ are derived in [9] and [10]:

  * •

All of the zeros of $\omega$ correspond to conical singularities of $g$. For
each zero of $\omega$ the corresponding singular angle is of the form
$2\pi\alpha$ where $\alpha$ is an integer and the order of $\omega$ at the
zero is $\alpha-1$. The function $K$ can be smoothly extended to these zeros,
where $dK$ vanishes. At each zero of $\omega$, the value of $K$ falls within
the range $[K_{1},K_{2}]$, hence we refer to these zeros as saddle points of
$K$.

  * •

$\omega$ possesses only simple poles, which correspond to smooth extremal
points of $K$ and singularities of $g$, excluding the zeros of $\omega$.
Specifically, these poles of $\omega$ consist of all of the global maximum
points and the minimum points of $K$. The residue of $\omega$ at each of these
poles is a real number.

  * •

If all singularities of $g$ are conical singularities, denote the constant
$-\frac{3}{(K_{1}-K_{2})(K_{2}+2K_{1})}$ by $\sigma$ and
$-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}$ by $\lambda$. Then at a maximum point of
$K$ the residue of $\omega$ is $\sigma\alpha$ if at this point $g$ has the
singular angle $2\pi\alpha$ or the residue of $\omega$ is $\sigma$ if this
maximum point of $K$ is the smooth point of $g$. At a minimum point of $K$ the
residue of $\omega$ is $\sigma\lambda\alpha$ if at this point $g$ has the
singular angle $2\pi\alpha$ or the residue of $\omega$ is $\sigma\lambda$ if
this minimum point of $K$ is the smooth point of $g$.

  * •

If the singularities of $g$ contain conical and cusp singularities, then each
cusp singularity of $g$ is a pole of $\omega$, the residue of $\omega$ at each
cusp singularity is positive and each cusp singularity is a minimum point of
$K$. Denote $K_{2}$ by $\mu$. Then if $p$ is a conical singularity with
singular angle $2\pi\alpha$ and is a pole of $\omega$, the residue of $\omega$
at $p$ is $-\alpha/3\mu^{2}$ and $\lim_{x\rightarrow p}K(x)=-2\mu$; if $e$ is
a smooth critical point of $K$, the residue of $\omega$ at $e$ is
$-1/3\mu^{2}$ and $\lim_{x\rightarrow e}K(x)=-2\mu$.

  * •

$\omega+\bar{\omega}$ is exact on
$\mathcal{M}\setminus\\{poles~{}of~{}\omega\\}$.

###  2.2 Reduce the existence of non-CSC HCMU metrics to the existence of some
kind of meromorphic 1-forms

In this subsection, we will review some results in [9] and [7]. For the
convenience of the reader, we will provide detailed proofs of the main
theorems. And we will review the energy integral formula for non-CSC HCMU
metrics in [22]. First, by virtue of a theorem from [25], we can establish the
following theorem.

######  Theorem 2.1 ([9]).

Let $\mathcal{M}$ be a Riemann surface, $p_{1},\ldots,p_{L}$ be $L(L\geq 2)$
points on $\mathcal{M}$ and $d_{1},\ldots,d_{L}$ be $L$ nonzero real numbers
with $d_{1}+\ldots+d_{L}=0$. Then there exists a meromorphic 1-form $\omega$
on $\mathcal{M}$ such that

  * 1)

$\omega$ only has $L$ simple poles at $p_{1},\ldots,p_{L}$ with
$Res_{p_{l}}(\omega)=d_{l},~{}l=1,\ldots,L$,

  * 2)

$\omega+\bar{\omega}$ is exact on
$\mathcal{M}\setminus\\{p_{1},\ldots,p_{L}\\}$.

Subsequently, Q. Chen and Y.Y. Wu [9] established the following theorem.

######  Theorem 2.2 ([9]).

Let $\mathcal{M}$ be a compact Riemann surface and $\omega$ be a meromorphic
1-form on $\mathcal{M}$ satisfying the conditions:

  * 1)

$\omega$ only has simple poles,

  * 2)

At each pole the residue of $\omega$ is a real number,

  * 3)

$\omega+\bar{\omega}$ is exact on $\mathcal{M}\setminus\\{\text{poles
of}~{}\omega\\}$.

Then, there exists a non-CSC HCMU metric with conical singularities such that
$\omega$ is the character 1-form of the metric.

###### Proof.

First by Theorem 2.1 a meromorphic 1-form on $\mathcal{M}$ satisfying the
conditions 1), 2), 3) in Theorem 2.2 always exists. Suppose
$p_{1},\ldots,p_{L}$ are the poles of $\omega$ in which $p_{1},\ldots,p_{J}$
are the poles where the residues of $\omega$ are negative and
$p_{J+1},\ldots,p_{L}$ are the poles where the residues of $\omega$ are
positive. Let $K_{1},K_{2}$ be two real numbers satisfying:

$K_{1}>0,~{}~{}K_{1}>K_{2}>-(K_{1}+K_{2}).$

Consider the following equation:

$\cfrac{dK}{-\frac{1}{3}(K-K_{1})(K-K_{2})(K+K_{1}+K_{2})}=\omega+\bar{\omega},~{}~{}\text{and}~{}K(p_{0})=K_{0},$
(6)

where $K_{2}<K_{0}<K_{1}$ and
$p_{0}\in\mathcal{M}\setminus\\{p_{1},\ldots,p_{L}\\}$. One can prove that
there exists a unique solution $K$ of (6) on $\mathcal{M}$ which satisfies
that $K$ is smooth on $\mathcal{M}\setminus\\{p_{1},\ldots,p_{L}\\}$ and is
continuous on $\mathcal{M}$. Then construct a metric

$g=-\frac{4}{3}(K-K_{1})(K-K_{2})(K+K_{1}+K_{2})\omega\bar{\omega}.$

One can prove $g$ is a non-CSC HCMU metric, $K$ is the Gauss curvature of $g$
with $K_{1},K_{2}$ being the maximum and the minimum of $K$ and $\omega$ is
the character 1-form of $g$. Therefore $g$ has the conical singularities at
the zeros and the poles of $\omega$. Denote
$-\frac{3}{(K_{1}-K_{2})(K_{2}+2K_{1})}$ by $\sigma$ and
$-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}$ by $\lambda$. At the zeros of $\omega$
the singular angles of $g$ are of the form $2\pi(ord_{p}(\omega)+1)$, and at
the poles of $\omega$ the singular angles of $g$ are of the form
$2\pi\frac{Res_{p}(\omega)}{\sigma}$ or
$2\pi\frac{Res_{p}(\omega)}{\lambda\sigma}$ depending on the sign of
$Res_{p}(\omega)$. $\frac{Res_{p}(\omega)}{\sigma}=1$ or
$\frac{Res_{p}(\omega)}{\lambda\sigma}=1$ means that $p$ is a smooth point of
$g$. ∎

In [10], Q. Chen, B. Xu and Y.Y. Wu expanded upon Theorem 2.2 as follows.

######  Theorem 2.3 ([10]).

Let $\mathcal{M}$ be a compact Riemann surface and $\omega$ be a meromorphic
1-form on $\mathcal{M}$ satisfying the conditions:

  * 1)

$\omega$ only has simple poles,

  * 2)

At each pole the residue of $\omega$ is a real number,

  * 3)

$\omega+\bar{\omega}$ is exact on $\mathcal{M}\setminus\\{\text{poles
of}~{}\omega\\}$.

Then, there exists a non-CSC HCMU metric with cusp singularities and conical
singularities such that $\omega$ is the character 1-form of the metric.

###### Proof.

Suppose $p_{1},\ldots,p_{L}$ are the poles of $\omega$ in which
$p_{1},\ldots,p_{J}$ are the poles where the residues of $\omega$ are negative
and $p_{J+1},\ldots,p_{L}$ are the poles where the residues of $\omega$ are
positive. Let $\mu<0$ be a constant. Consider the following equation:

$\cfrac{dK}{-\frac{1}{3}(K-\mu)^{2}(K+2\mu)}=\omega+\bar{\omega},~{}~{}\text{and}~{}K(p_{0})=K_{0},$
(7)

where $\mu<K_{0}<-2\mu$ and
$p_{0}\in\mathcal{M}\setminus\\{p_{1},\ldots,p_{L}\\}$. One can prove that
there exists a unique solution $K$ of (7) on $\mathcal{M}$ which satisfies
that $K$ is smooth on $M\setminus\\{p_{1},\ldots,p_{L}\\}$ and is continuous
on $\mathcal{M}$. Then construct a metric

$g=-\frac{4}{3}(K-\mu)^{2}(K+2\mu)\omega\bar{\omega}.$

One can prove $g$ is a non-CSC HCMU metric, $K$ is the Gauss curvature of $g$
with $-2\mu,\mu$ being the maximum and the minimum of $K$ and $\omega$ is the
character 1-form of $g$. Therefore $g$ has the conical singularities at the
zeros and the poles with negative residues of $\omega$, and cups singularities
at the poles with positive residues of $\omega$. At the zeros of $\omega$ the
singular angles of $g$ are of the form $2\pi(ord_{p}(\omega)+1)$, and at the
poles with negative residues of $\omega$ the singular angles of $g$ are of the
form $2\pi(-3\mu^{2}Res_{p}(\omega))$. $-3\mu^{2}Res_{p}(\omega)=1$ means that
$p$ is a smooth point of $g$. ∎

To construct a prescribed non-CSC HCMU metric, Theorem 2.2 or 2.3 assert that
it suffices to find a suitable meromorphic 1-form meeting the criteria.
However, finding such a 1-form can be challenging due to the unknown smooth
points in the metric’s singularities and the need to identify which points
among the given ones are zeros of the 1-form. It’s worth noting that a
meromorphic 1-form on the Riemann sphere $S^{2}$ that adheres to the
conditions 1) and 2) in Theorem 2.2 or 2.3 automatically fulfills condition 3)
(cf. [9]). By this fact, Y.Y. Wu and the second author [23] established the
following theorem.

######  Theorem 2.4 ([23]).

Let $p_{1},\ldots,p_{N}$ be $N(N\geq 3)$ points on $S^{2}$ and
$2\pi\alpha_{1},\ldots,2\pi\alpha_{N}$ be $N$ positive real numbers with
$\alpha_{n}\neq 1$, for $n=1,2,\ldots,N$. If at least $N-2$ of the
$\alpha_{1},\ldots,\alpha_{N}$ are integers, then there exists a non-CSC HCMU
metric which has conical singularities $p_{1},\ldots,p_{N}$ with singular
angles $2\pi\alpha_{1},\ldots,2\pi\alpha_{N}$ respectively.

At last, we introduce the energy integral formula for non-CSC HCMU metrics,
which was proved by using Stokes’ formula in [22] and will be used in the
proof of our classification.

######  Theorem 2.5 ([22]).

Suppose $\mathcal{M}$ is a compact Riemann surface, $p_{1},\ldots,p_{N}$ are
$N$ points on $\mathcal{M}$, and $g$ is a non-CSC HCMU metric on $\mathcal{M}$
with singular points $p_{1},\ldots,p_{N}$. Suppose $K$ is the Gaussian
curvature of $g$,
$\\{q_{1},\ldots,q_{s}\\}\subseteq\mathcal{M}\setminus\\{p_{1},\ldots,p_{N}\\}$
is the set of smooth extremal points of $K$, and $\omega$ is the character
1-form of $g$. Set
$\mathcal{M}^{\prime}=\mathcal{M}\setminus\\{p_{1},\ldots,p_{N},q_{1},\ldots,q_{s}\\}$,
and

$\mathcal{C}_{n}(g)=\int_{\mathcal{M}^{\prime}}K^{n}dg,~{}n=0,1,2,\ldots,$

which are called the $n$-th energy integral. Then

$\mathcal{C}_{n}(g)=\frac{6\alpha_{max}(K_{1}^{n+1}-K_{2}^{n+1})}{(n+1)(K_{1}-K_{2})(K_{2}+2K_{1})},$

where $\alpha_{max}$ is the sum of conical angles at the maximum points of
$K$, and $K_{1},K_{2}$ are the maximum and minimum of $K$, respectively.

###  2.3 One existence theorem for rational functions on the Riemann sphere

In this subsection, we will review an existence result for rational functions
on the Riemann sphere, as presented in [18]. For more results, we refer the
reader to the references cited in [18].

Let $X$ and $Y$ be two compact, connected Riemann surfaces, and consider a
holomorphic branched covering $f:X\rightarrow Y$ of degree $d$. At each point
$q$ in $Y$, there is partition $\lambda(q)=(k_{1},\ldots,k_{r})$ of $d$ that
characterizes the local behavior of $f$ near $q$. Over a suitable neighborhood
of $q$ in $Y$, the map $f$ is equivalent to the map

$\\{1,\ldots,r\\}\times\mathbb{D}\rightarrow\mathbb{D},~{}~{}(j,z)\mapsto
z^{k_{j}},~{}~{}\text{where}~{}\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\},$

with $q$ corresponding to $0$ in $\mathbb{D}$. For any partition
$\lambda=(k_{1},\ldots,k_{r})$ of $d$, we define its length $Len(\lambda)=r$.
We refer to a partition $\lambda$ of $d$ as non-trivial if $Len(\lambda)<d$.
For the branched covering $f:X\rightarrow Y$, we call a point $q$ in $Y$ a
branch point of $f$ if and only if $\lambda(q)$ is non-trivial, and we call
the set of branch points of $f$ the branch set of $f$, denoted by $B_{f}$. The
collection $\Lambda=\\{\lambda(q):q\in B_{f}\\}$ (with repetitions allowed) is
called the branch data of $f$ and

$v(f):=\sum_{q\in B_{f}}(d-Len(\lambda(q)))$

the total branching data of $f$. By the Riemann-Hurwitz formula, we have

$v(f)=2g(X)-2-d(2g(Y)-2),$

where $g(X)$ (resp. $g(Y)$) denotes the genus of $X$ (resp. $Y$). Therefore,
the total branching order $v(f)$ is an even non-negative integer.

A well-known realizability problem which arises in topology asks whether,
given a compact connected Riemann surface $Y$ and a collection
$\Lambda=\\{\lambda_{1},\ldots,\lambda_{k}\\}$ of non-trivial partitions of a
positive integer $d$, there exist another compact connected Riemann surface
$X$ together with a branched covering $f:X\rightarrow Y$ such that $\Lambda$
is its branch data. If such an $X$ and $f$ exist, we say that $\Lambda$ is
realizable or realized by a branched covering. Boccara [2] obtained the
following theorem.

######  Theorem 2.6 ([2]).

Suppose

$\Lambda=\\{(a_{1},\ldots,a_{p}),(b_{1},\ldots,b_{q}),(m+1,1,\ldots,1)\\}$

is a collection of a partition of a positive integer $d$. Then there exists a
branched covering of $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ with
$\Lambda$ being its branch data if and only if it satisfies one of the
following:
(i) $v(\Lambda)\geq 2d$ is even.
(ii) $v(\Lambda)=2d-2$ and $m<d/{\rm
GCD}(a_{1},\ldots,a_{p},b_{1},\ldots,b_{q})$

Recently, J.J. Song and B. Xu [18] generalized the second part of Boccara’s
result.

######  Theorem 2.7 ([18]).

Let $d$ and $l$ be two positive integers. Consider a collection

$\Lambda=\\{(a_{1},\ldots,a_{p}),(b_{1},\ldots,b_{q}),(m_{1}+1,1,\ldots,1),\ldots,(m_{l}+1,1,\ldots,1)\\}$

of $l+2$ partitions of $d$ where $(m_{1},\ldots,m_{l})$ is a partition of
$p+q-2>0$. Then there exists a rational function on
$\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ with $\Lambda$ being its
branch data if and only if

$max(m_{1},\ldots,m_{l})<d/{\rm GCD}(a_{1},\ldots,a_{p},b_{1},\ldots,b_{q}).$

##  3 Classification of non-CSC HCMU metrics on $S^{2}_{\\{\alpha\\}}$

In this section, we will give the proof Theorem 1.5. Our strategy involves a
case-by-case analysis.

If $\alpha=0$, according to Theorem 1.1, a non-CSC HCMU metric $g$ exists on
$S^{2}_{\\{\alpha\\}}$ if and only if its single singularity is a cusp.
Furthermore, such a metric is uniquely determined by its total area and is
necessarily rotationally symmetric. Therefore, to classify non-CSC HCMU
metrics on $S^{2}_{\\{\alpha\\}}$, we need only consider the case where the
singularity is conical.

(A) For $2\leq\alpha\in\mathbb{Z}$, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha\\}}$ can exist under the following two conditions.

(A-1) The singularity is an extremal point of the Gaussian curvature $K$ of
$g$.

Since $\alpha>1$, the singularity is a maximum point of $K$. Denote the
maximum and minimum values of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has a single
singularity and the singularity is a maximum of $K$, $\omega$ has two simple
poles. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$
and $\infty$ are poles of $\omega$, with $\infty$ representing the conical
singularity. Then
$Res_{0}(\omega)=\sigma\lambda,Res_{\infty}(\omega)=\alpha\sigma$. By the
Residue theorem, we obtain
$\sigma\lambda+\alpha\sigma=\sigma(\lambda+\alpha)=0$, which implies
$\alpha=-\lambda=\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}$. Thus

$\omega=-\frac{\sigma\alpha}{z}dz.$

Conversely, by Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha\\}}$ such that the singularity is the maximum point of the
Gaussian curvature $K$ of $g$. Furthermore, by Theorem 2.5, one can easily
check that $g$ is uniquely determined by the total area and $\alpha$, and $g$
must be rotationally symmetric.

(A-2) The singularity is the saddle point of the Gaussian curvature $K$ of
$g$.

Denote the maximum and minimum values of $K$ by $K_{1}$ and $K_{2}$
respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has a single
singularity and the singularity is the saddle point of $K$, $\omega$ has a
simple zero of order $\alpha-1$ and $\alpha+1$ simple poles which are smooth
extremal points of $K$. Suppose $K$ has $I_{1}$ maximum points and $I_{2}$
minimum points, then $I_{1}+I_{2}=\alpha+1$. By
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}<-1$, we obtain $I_{1}>I_{2}$.

Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$ is the
zero of $\omega$, and that $a_{1},\ldots,a_{I_{1}}$ are poles of $\omega$ with
negative residues, while $b_{1},\ldots,b_{I_{2}}$ are poles of $\omega$ with
positive residues. Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1}~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2}$.
From these, we derive
$I_{1}\sigma+I_{2}\sigma\lambda=\sigma(I_{1}+I_{2}\lambda)=0$, which leads to
$\lambda=-\frac{I_{1}}{I_{2}}$. Thus there is a nonzero complex number
$\widetilde{B}$ such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

######  Proposition 3.1.

Given an integer $\alpha\geq 2$, let $I_{1}$ and $I_{2}$ be two positive
integers satisfying $I_{1}+I_{2}=\alpha+1$ and $I_{1}>I_{2}$. Then, there
exists a meromorphic 1-form $\omega$ on $S^{2}=\mathbb{C}\cup\\{\infty\\}$,
defined by the form

$\omega=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant, and
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers, if and only if one of the following two
conditions holds.
(1) $I_{2}=1$;
(2) $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$.

###### Proof.

Sufficiency

If $I_{2}=1$, a direct calculation verifies the result.

For $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$, let $(I_{1},I_{2})=m\geq 1$. By
Theorem 2.6, there exists a meromorphic function
$f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$ of degree
$\frac{I_{1}I_{2}}{m}$ and with branch data

$\Lambda=\\{(\underbrace{\frac{I_{2}}{m},\ldots,\frac{I_{2}}{m}}_{I_{1}}),(\underbrace{\frac{I_{1}}{m},\ldots,\frac{I_{1}}{m}}_{I_{2}}),(\alpha,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}}{m}}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{m}}},$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $C\neq 0$ is a constant. Then

$\frac{df}{f}=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\frac{I_{2}}{m}(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$

where $B\neq 0$ is a constant, is the desired meromorphic 1-form.

Necessity

Suppose there exists a meromorphic 1-form on
$S^{2}=\mathbb{C}\cup\\{\infty\\}$, defined as follows

$\omega=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers.

If $I_{2}=1$, there is no further proof required. Consequently, we only need
to consider the case (2).

Suppose $I_{2}\geq 2$ and $I_{2}|I_{1}$, then the function

$f(z)=exp(\int\omega)=C\frac{\prod_{k=1}^{I_{1}}(z-a_{k})}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}}}$

is a meromorphic function with degree $I_{1}$, where $C\neq 0$ is a constant.
The derivative of $f$ is

$f^{\prime}(z)=\frac{CBz^{\alpha-1}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}+1}}.$

This implies $\alpha\leq I_{1}$. Consequently, $I_{1}=\alpha$ and $I_{2}=1$.
It is a contradiction. So $I_{2}\nmid I_{1}$. ∎

By Proposition 3.1 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha\\}}$ such that the singularity is a saddle point of the
Gaussian curvature $K$ of $g$. Furthermore, $K$ possesses $I_{1}$ maximum
points and $I_{2}$ minimum points, where $I_{1}=\alpha,I_{2}=1$ or $I_{2}\geq
2$ and $I_{2}\nmid I_{1}$, with $I_{1}>I_{2}$ and their sum satisfying
$I_{1}+I_{2}=\alpha+1$. Since the number of saddle and extremal points of $K$
exceeds $3$, by virtue of Theorem 2.5, it is straightforward to ascertain that
$g$ is uniquely determined by the total area, $\frac{I_{1}}{I_{2}}$, $\alpha$
and the initial value $K(z_{0})=K_{0}$.

(B) When $\alpha\notin\mathbb{Z}^{+}$, a non-CSC HCMU metric $g$ can exist on
$S^{2}_{\\{\alpha\\}}$ under the following two conditions.

(B-1) The singularity is the maximum point of the Gaussian curvature $K$ of
$g$.

First we obtain $\alpha>1$. Denote the maximum and minimum values of $K$ by
$K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has a single
singularity and the singularity is a maximum of $K$, $\omega$ has two simple
poles. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$
and $\infty$ are simple poles of $\omega$, with $\infty$ being the conical
singularity. Then $Res_{0}(\omega)=\sigma\lambda$ and
$Res_{\infty}(\omega)=\alpha\sigma$. From this, we deduce that
$\sigma\lambda+\alpha\sigma=\sigma(\lambda+\alpha)=0$, which implies
$\alpha=-\lambda=\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}$. Thus

$\omega=-\frac{\sigma\alpha}{z}dz.$

Conversely, by Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha\\}}$ such that the singularity is the maximum point of the
Gaussian curvature $K$ of $g$. Furthermore, by Theorem 2.5, one can easily
check that $g$ is uniquely determined by the total area and $\alpha$, and must
be rotationally symmetric.

(B-2) The singularity is the minimum point of the Gaussian curvature $K$ of
$g$.

First we obtain $\alpha<1$. Denote the maximum and minimum values of $K$ by
$K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has a single
singularity and the singularity is a minimum of $K$, $\omega$ has two simple
poles. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$
and $\infty$ are simple poles of $\omega$, with $\infty$ being the conical
singularity. Then $Res_{0}(\omega)=\sigma$ and
$Res_{\infty}(\omega)=\alpha\sigma\lambda$. From this, we deduce that
$\sigma+\alpha\sigma\lambda=\sigma(1+\alpha\lambda)=0$, which implies
$\alpha=-1/\lambda=\frac{2K_{2}+K_{1}}{2K_{1}+K_{2}}$. Thus

$\omega=-\frac{\sigma}{z}dz.$

Conversely, by Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha\\}}$ such that the singularity is the minimum point of the
Gaussian curvature $K$ of $g$. Furthermore, by Theorem 2.5, one can easily
check that $g$ is uniquely determined by the total area and $\alpha$, and must
be rotationally symmetric.

##  4 Classification of non-CSC HCMU metrics on $S^{2}_{\\{\alpha,\beta\\}}$

In this section, we will give the proof the proof Theorem 1.6. Our strategy is
the same as the proof of Theorem 1.5.

By Theorem 1.1, if $\alpha=\beta=0$, there is no non-CSC HCMU metric on
$S^{2}_{\\{\alpha,\beta\\}}$. Consequently, at most one of the parameters
$\alpha$ and $\beta$ can be equal to $0$.

Firstly, if there is a zero in $\alpha,\beta$, without loss of generality, we
can suppose $\beta=0$. Then, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exists if and only if $g$ possesses both a
conical singularity and a cusp singularity. It’s worth noting that G.F. Wang
and X.H. Zhu have classified the case where $\alpha<\frac{1}{2}$, as stated in
Theorem 1.2. Now, we proceed to classify non-CSC HCMU metrics for all
$0<\alpha\neq 1$.

(A) If $2\leq\alpha\in\mathbb{Z}$, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exists if and only if one of the following two
cases holds.

(A-1) The singularity of conical angle $2\pi\alpha$ is the maximum point of
the Gaussian curvature $K$ of $g$.

Obviously, in this scenario, $K$ precisely possesses exactly two extremal
points. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that the
conical and cusp singularities of $g$ are $0$ and $\infty$ respectively.
Denote the minimum of $K$ by $\mu<0$. Then the character 1-form of $g$ is
$\omega=-\frac{\alpha}{3\mu^{2}}\cdot\frac{1}{z}dz$.

Conversely, by Theorem 2.3, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the maximum point of the Gaussian curvature $K$ of $g$ and
another singularity is the minimum point of $K$. Furthermore, by Theorem 2.5,
one can easily check that $g$ is uniquely determined by the total area and
$\alpha$, and must be rotationally symmetric.

(A-2) The singularity of conical angle $2\pi\alpha$ is the saddle point of the
Gaussian curvature $K$ of $g$.

In this case, $K$ possesses exactly $\alpha$ smooth maximum points. Regard
$S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that the conical and
cusp singularities of $g$ are $0$ and $\infty$ respectively, and the smooth
maximum points of $K$ are $a_{1},\ldots,a_{\alpha}$. Denote the minimum of $K$
by $\mu<0$. Then the character 1-form of $g$ can be written as

$\omega=-\frac{1}{3\mu^{2}}(\sum_{k=1}^{\alpha}\frac{1}{z-a_{k}})dz=\frac{\widetilde{B}z^{\alpha-1}}{\prod_{k=1}^{\alpha}(z-a_{k})}dz,$

where $\widetilde{B}\neq 0$ is a constant.

Conversely, it is straightforward to establish the following proposition.

######  Proposition 4.1.

Let $\alpha\geq 2$ be an integer. For any real number $\mu<0$, there exists a
meromorphic 1-form $\omega$ on $S^{2}=\mathbb{C}\cup\\{\infty\\}$, defined by
the form

$\omega=-\frac{1}{3\mu^{2}}(\sum_{k=1}^{\alpha}\frac{1}{z-a_{k}})dz=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{\alpha}(z-a_{k})}dz,$
(8)

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{\alpha}\in\mathbb{C}\setminus\\{0\\}$ are distinct complex
numbers.

By Proposition 4.1 and Theorem 2.3, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the saddle point of the Gaussian curvature $K$ of $g$ and
another singularity is the minimum point of $K$. Additionally, by Theorem 2.5,
one can easily check that $g$ is uniquely determined by the total area and
$\alpha$, and the initial value $K(z_{0})=K_{0}$.

(B) If $\alpha\notin\mathbb{Z}$, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exists if and only if the Gaussian curvature $K$
of $g$ has both a maximum and a minimum points.

Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can assume that the conical
and cusp singularities of $g$ are $0$ and $\infty$ respectively. Denote the
minimum of $K$ by $\mu<0$. Then the character 1-form of $g$ is
$\omega=-\frac{\alpha}{3\mu^{2}}\cdot\frac{1}{z}dz$.

Conversely, by Theorem 2.3, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the maximum point of the Gaussian curvature $K$ of $g$ and
another singularity is the minimum point of $K$. Furthermore, by Theorem 2.5,
one can easily check that $g$ is uniquely determined by the total area and
$\alpha$, and must be rotationally symmetric.

Secondly, if $\alpha,\beta\notin\mathbb{Z}$, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exists if and only if $\alpha\neq\beta$.
Furthermore, $g$ must be a football. This classification is due to X.X. Chen,
as stated in Theorem 1.4.

Thirdly, if there is only one positive integer in $\alpha,\beta$, without loss
of generality, we can assume that $\alpha\in\mathbb{Z}^{+}$ and
$0<\beta\notin\mathbb{Z}$. Then, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exits if and only if one of the following two
conditions holds.

(A) If the singularity of conical angle $2\pi\alpha$ is an extremal point of
the Gaussian curvature $K$ of $g$, it leads to two distinct scenarios.

(A-1) If the singularity of conical angle $2\pi\alpha$ is the maximum point of
the Gaussian curvature $K$ of $g$, then another singularity is the minimum
point of $K$ and $\alpha>\beta$. Moreover, $g$ must be a football.

(A-2) If the singularity of conical angle $2\pi\alpha$ is the minimum point of
the Gaussian curvature $K$ of $g$, then another singularity is the maximum
point of $K$ and $\alpha<\beta$. Moreover, $g$ must be a football.

(B) If the singularity of conical angle $2\pi\alpha$ is the saddle point of
the Gaussian curvature $K$ of $g$, it leads to two distinct scenarios.

(B-1) If the singularity of conical angle $2\pi\beta$ is a maximum point of
the Gaussian curvature $K$ of $g$, then $K$ has exactly $\alpha$ smooth
extremal points. Suppose $K$ has $I_{1}\geq 0$ smooth maximum points and
$I_{2}\geq 1$ smooth minimum points, then $I_{1}+I_{2}=\alpha$. Denote the
maximum and minimum values of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has two
singularities and one is the saddle point of $K$ and another is a maximum
point of $K$, $\omega$ has a zero of order $\alpha-1$, $\alpha$ simple poles
which are smooth extremal points of $K$ and a simple pole which is the
singularity of singular angle $2\pi\beta$. Regard $S^{2}$ as
$\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$ is the zero of $\omega$,
$a_{1},\ldots,a_{I_{1}},a_{I_{1}+1}$ are poles of $\omega$ at which residues
are negative and $b_{1},\ldots,b_{I_{2}}$ are poles of $\omega$ at which
residues are positive. Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1},Res_{a_{I_{1}+1}}(\omega)=\sigma\beta~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2}$.
From this, we derive
$I_{1}\sigma+\sigma\beta+I_{2}\sigma\lambda=\sigma(I_{1}+\beta+I_{2}\lambda)=0$,
which implies $\lambda=-\frac{I_{1}+\beta}{I_{2}}$. By
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}=-\frac{I_{1}+\beta}{I_{2}}<-1$, we
obtain $I_{1}+\beta>I_{2}$. Thus there is a nonzero $\widetilde{B}$ such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}}{\prod_{k=1}^{I_{1}+1}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

Similar to the proof of Proposition 4.5, one can establish the following
proposition.

######  Proposition 4.2.

Let $\alpha\geq 2,I_{1}\geq 0$ and $I_{2}\geq 1$ be integers such that
$I_{1}+I_{2}=\alpha$. Additionally, let $0<\beta\notin\mathbb{Z}$ satisfy the
condition $I_{1}+\beta>I_{2}$. Then there exists a meromorphic 1-form $\omega$
on $S^{2}=\mathbb{C}\cup\\{\infty\\}$, defined by the form

$\omega=\frac{Bz^{\alpha-1}}{(z-a_{I_{1}+1})\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},a_{I_{1}+1},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0,1\\}$
are distinct complex numbers.

By Proposition 4.2 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the saddle points of the Gaussian curvature $K$ and the
singularity of singular angle $2\pi\beta$ is a maximum point of $K$.

(B-2) If the singularity of conical angle $2\pi\beta$ is the minimum point of
the Gaussian curvature $K$ of $g$, then $K$ has exactly $\alpha$ smooth
extremal points. Suppose $K$ has $I_{1}\geq 1$ smooth maximum points and
$I_{2}\geq 0$ smooth minimum points, then $I_{1}+I_{2}=\alpha$. Denote the
maximum and minimum of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has two
singularities and one is the saddle point of $K$ and another is a maximum
point of $K$, $\omega$ has a zero of order $\alpha-1$, $\alpha$ simple poles
which are smooth extremal points of $K$ and a simple pole which is the
singularity of singular angle $2\pi\beta$. Regard $S^{2}$ as
$\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$ is the zero of $\omega$,
$a_{1},\ldots,a_{I_{1}}$ are poles of $\omega$ at which residues are negative
and $b_{1},\ldots,b_{I_{2}},b_{I_{2}+1}$ are poles of $\omega$ at which
residues are positive. Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1}~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2},Res_{b_{I_{2}+1}}(\omega)=\sigma\lambda\beta$.
From these, we derive
$I_{1}\sigma+\sigma\lambda\beta+I_{2}\sigma\lambda=\sigma(I_{1}+\lambda\beta+I_{2}\lambda)=0$,
which implies $\lambda=-\frac{I_{1}}{I_{2}+\beta}$. By
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}=-\frac{I_{1}}{I_{2}+\beta}<-1$, we
obtain $I_{1}>I_{2}+\beta$. Consequently, there exists a nonzero
$\widetilde{B}$ such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{I_{1}\beta}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

Similar to the proof of Proposition 4.6, we can establish the following
proposition.

######  Proposition 4.3.

Let $\alpha\geq 2,I_{1}\geq 1$ and $I_{2}\geq 0$ be integers such that
$I_{1}+I_{2}=\alpha$. Additionally, let $0<\beta\notin\mathbb{Z}$ satisfy
$I_{1}>I_{2}+\beta$. Then there exists a meromorphic 1-form $\omega$ on
$S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by the form

$\omega=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{I_{1}\beta}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},a_{I_{1}},b_{1},\ldots,b_{I_{2}+1}\in\mathbb{C}\setminus\\{0,1\\}$
are distinct complex numbers.

By Proposition 4.3 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the saddle points of the Gaussian curvature $K$ and the
singularity of singular angle $2\pi\beta$ is a minimum point of $K$.

Fourthly, if $\alpha,\beta\in\mathbb{Z}^{+}$, a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ exists if and only if one of the following two
conditions holds.

(A) The two singularities are both saddle points of the Gaussian curvature $K$
of $g$.

Denote the maximum and minimum of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Since $g$ has exactly two
singularities and the singularities are both saddle points of the Gauss
curvature, $\omega$ has two zeros of orders $\alpha-1$ and $\beta-1$
respectively, and $\alpha+\beta$ simple poles which are smooth extremal points
of $K$. Suppose $K$ has $I_{1}$ maximum points and $I_{2}$ minimum points,
then $I_{1}+I_{2}=\alpha+\beta$. Regard $S^{2}$ as
$\mathbb{C}\cup\\{\infty\\}$. We can assume that $0$ and $1$ are the zeros of
$\omega$, $a_{1},\ldots,a_{I_{1}}$ are poles of $\omega$ at which residues are
negative and $b_{1},\ldots,b_{I_{2}}$ are poles of $\omega$ at which residues
are positive. Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1}~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2}$.
Applying the Residue theorem, we obtain
$I_{1}\sigma+I_{2}\sigma\lambda=\sigma(I_{1}+I_{2}\lambda)=0$, which implies
$\lambda=-\frac{I_{1}}{I_{2}}$. Since
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}=-\frac{I_{1}}{I_{2}}<-1$, we
obtain $I_{1}>I_{2}$. Consequently, there is a nonzero complex number
$\widetilde{B}$ such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}(z-1)^{\beta-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

######  Proposition 4.4.

Let $\alpha\geq 2,\beta\geq 2,I_{1}\geq 2$ and $I_{2}\geq 1$ be 4 integers
such that $I_{1}>I_{2}$ and $I_{1}+I_{2}=\alpha+\beta$. Then, there exists a
meromorphic 1-form $\omega$ on $S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by
the form

$\omega=\frac{Bz^{\alpha-1}(z-1)^{\beta-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$
(9)

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant, and
$a_{1},\ldots,a_{I},b_{1},\ldots,b_{I}\in\mathbb{C}\setminus\\{0,1\\}$ are
distinct complex numbers, if and only if one of the following conditions
holds.
(1) $I_{2}=1$;
(2) $I_{2}\geq 2,I_{2}\mid I_{1}$ and $I_{1}\geq\alpha$;
(3) $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$.

###### Proof.

Sufficiency

If $I_{2}=1$, a direct calculation verifies the result.

For $I_{2}\geq 2,I_{2}\mid I_{1}$ and $I_{1}\geq\alpha$, by Theorem 2.6, there
exists a meromorphic function
$f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$ of degree
$I_{1}$ with branch data

$\Lambda=\\{(\underbrace{\frac{I_{1}}{I_{2}},\ldots,\frac{I_{1}}{I_{2}}}_{I_{2}}),(\alpha,1,\ldots,1),(\beta,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}}}$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $C\neq 0$ is a constant. Then

$\frac{df}{f}=\frac{Bz^{\alpha-1}(z-1)^{\beta-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\neq 0$ is a constant, is the desired meromorphic 1-form.

For $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$, set $(I_{1},I_{2})=m\geq 1$. Then,
by Theorem 2.7, there exists a meromorphic function
$f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$ of degree
$\frac{I_{1}I_{2}}{m}$ and with branch data

$\Lambda=\\{(\underbrace{\frac{I_{2}}{m},\ldots,\frac{I_{2}}{m}}_{I_{1}}),(\underbrace{\frac{I_{1}}{m},\ldots,\frac{I_{1}}{m}}_{I_{2}}),(\alpha,1,\ldots,1),(\beta,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}}{m}}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{m}}}$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $C\neq 0$ is a constant. Then

$\frac{df}{f}=\frac{Bz^{\alpha-1}(z-1)^{\beta-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\frac{I_{2}}{m}(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\neq 0$ is a constant. Thus $\frac{m}{I_{2}}\cdot\frac{df}{f}$ is the
desired meromorphic 1-form.

Necessity

Suppose there exists a meromorphic 1-form on
$S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by the form

$\omega=\frac{Bz^{\alpha-1}(z-1)^{\beta-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{I_{1}}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers.

If $I_{2}=1$ or $I_{2}\geq 2$ and $I_{2}\nmid I_{1}$, there is no further
proof required. Consequently, we only need to consider the case (2).

Suppose $I_{2}\geq 2$ and $I_{2}|I_{1}$. Then

$f(z)=exp(\int\omega)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}}}$

is a meromorphic function on $\mathbb{C}\cup\\{\infty\\}$ with degree $I_{1}$,
where $C\neq 0$ is a constant. The derivative of $f$ is

$f^{\prime}(z)=\frac{CBz^{\alpha-1}(z-1)^{\beta-1}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}+1}}.$

Then $\alpha\leq I_{1}$.

The proofs of other two cases are similar. ∎

By Proposition 4.4 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity are all saddle points
of the Gaussian curvature $K$ of $g$. Furthermore, $K$ has $I_{1}$ maximum
points and $I_{2}$ minimum points.

(B) If one singularity is the saddle point of the Gaussian curvature $K$ of
$g$ and another singularity is an extremal point of $K$, it leads to two
distinct scenarios.

(B-1) Another singularity is a maximal point of $K$.

Denote the maximum and minimum of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Without loss of generality,
suppose the singularity of singular angle $2\pi\alpha$ is the saddle point of
$K$. Since $g$ has two singularities and one is the saddle point of $K$ and
another is a maximum point of $K$, $\omega$ has a zero of order $\alpha-1$,
$\alpha$ simple poles which are smooth extremal points of $K$ and a simple
pole which is the singularity of conical angle $2\pi\beta$. Suppose $K$ has
$I_{1}+1$ maximum points and $I_{2}$ minimum points, then
$I_{1}+I_{2}=\alpha$. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can
assume that $0$ is the zero of $\omega$, $a_{1},\ldots,a_{I_{1}},a_{I_{1}+1}$
are poles of $\omega$ at which residues are negative and
$b_{1},\ldots,b_{I_{2}}$ are poles of $\omega$ at which residues are positive.
Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1},Res_{a_{I_{1}+1}}(\omega)=\sigma\beta~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2}$.
This leads to
$I_{1}\sigma+\sigma\beta+I_{2}\sigma\lambda=\sigma(I_{1}+\beta+I_{2}\lambda)=0$,
which implies $\lambda=-\frac{I_{1}+\beta}{I_{2}}$. Since
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}=-\frac{I_{1}+\beta}{I_{2}}<-1$, we
obtain $I_{1}+\beta>I_{2}$. Consequently, there is a nonzero $\widetilde{B}$
such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}}{(z-a_{I_{1}+1})\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

######  Proposition 4.5.

Let $\alpha,\beta\geq 2,I_{1}\geq 0$ and $I_{2}\geq 1$ be $4$ integers such
that $I_{1}+I_{2}=\alpha$ and $I_{1}+\beta>I_{2}$. Then, there exists a
meromorphic 1-form $\omega$ on $S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by
the form

$\omega=\frac{Bz^{\alpha-1}}{(z-a_{I_{1}+1})\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$
(10)

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},a_{I_{1}+1},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0,1\\}$
are distinct complex numbers, if and only if one of the following conditions
holds.
(1) $I_{2}=1$;
(2) $I_{2}\geq 2,I_{2}\mid(I_{1}+\beta)$ and $I_{1}+\beta>\alpha-1$;
(3) $I_{2}\geq 2$ and $I_{2}\nmid(I_{1}+\beta)$.

###### Proof.

Sufficiency

If $I_{2}=1$, a direct calculation verifies the result.

For $I_{2}\geq 2,I_{2}\mid(I_{1}+\beta)$ and $I_{1}+\beta>\alpha-1$, by the
Theorem 2.6, there exists a meromorphic function
$f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$ of degree
$I_{1}+\beta$ and with branch data

$\Lambda=\\{(\underbrace{\frac{I_{1}+\beta}{I_{2}},\ldots,\frac{I_{1}+\beta}{I_{2}}}_{I_{2}}),(\beta,\underbrace{1,\ldots,1}_{I_{1}}),(\alpha,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C(z-a_{I_{1}+1})^{\beta}\prod_{k=1}^{I_{1}}(z-a_{k})}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}+\beta}{I_{2}}}}$

where
$a_{1},\ldots,a_{I_{1}},a_{I_{1}+1},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $C\neq 0$ is a constant. Then

$\frac{df}{f}=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\neq 0$ is a constant, is the desired meromorphic 1-form.

For $I_{2}\geq 2$ and $I_{2}\nmid(I_{1}+\beta)$, set
$(I_{1}+\beta,I_{2})=m\geq 1$, then $\frac{(I_{1}+\beta)I_{2}}{m}\geq
2(I_{1}+\beta)>I_{1}+I_{2}=\alpha$. By Theorem 2.6, there exists a meromorphic
function $f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$
of degree $\frac{(I_{1}+\beta)I_{2}}{m}$ and with branch data

$\Lambda=\\{(\frac{\beta
I_{2}}{m},\underbrace{\frac{I_{2}}{m},\ldots,\frac{I_{2}}{m}}_{I_{1}}),(\underbrace{\frac{I_{1}+\beta}{m},\ldots,\frac{I_{1}+\beta}{m}}_{I_{2}}),(\alpha,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C(z-a_{I_{1}+1})^{\frac{\beta
I_{2}}{m}}\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}}{m}}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}+\beta}{m}}}$

where
$a_{1},\ldots,a_{I_{1}},a_{I_{1}+1},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are disitnct complex numbers and $C\neq 0$ is a constant. Then

$\frac{df}{f}=\frac{Bz^{\alpha-1}}{(z-a_{I_{1}+1})\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=\frac{I_{2}}{m}(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{2}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\neq 0$ is a constant. Thus $\frac{m}{I_{2}}\cdot\frac{df}{f}$ is the
desired meromorphic 1-form.

Necessity

Suppose there exists a meromorphic 1-form on
$S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by the form

$\omega=\frac{Bz^{\alpha-1}}{(z-a_{I_{1}+1})\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}+\frac{\beta}{z-a_{I_{1}+1}}-\frac{I_{1}+\beta}{I_{2}}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $B\in\mathbb{C}\setminus\\{0\\}$ is a
constant.

If $I_{2}=1$ or $I_{2}\geq 2$ and $I_{2}\nmid(I_{1}+\beta)$, there is no
further proof required. Consequently, we only need to consider the case (2).

Suppose $I_{2}\geq 2$ and $I_{2}|(I_{1}+\beta)$. Then

$f(z)=exp(\int\omega)=\frac{C(z-a_{I_{1}+1})^{\beta}\prod_{k=1}^{I_{1}}(z-a_{k})}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}+\beta}{I_{2}}}}$

is a meromorphic function on $\mathbb{C}\cup\\{\infty\\}$ with degree
$I_{1}+\beta$, where $C\neq 0$ is a constant. The derivative of $f$ is

$f^{\prime}(z)=\frac{CBz^{\alpha-1}(z-a_{I_{1}+1})^{\beta-1}}{\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}}+1}}.$

Then $I_{1}+\beta>\alpha-1$. ∎

By Proposition 4.5 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the saddle points of the Gaussian curvature $K$ and the
singularity of singular angle $2\pi\beta$ is a maximum point of $K$.

(B-2) Another singularity is a minimal point of $K$.

Denote the maximum and minimum of $K$ by $K_{1}$ and $K_{2}$ respectively. Set

$\sigma=-\frac{1}{(K_{1}-K_{2})(K_{2}+2K_{1})},~{}~{}~{}\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}.$

Suppose the character 1-form of $g$ is $\omega$. Without loss of generality,
suppose the singularity of singular angle $2\pi\alpha$ is the saddle point of
$K$. Since $g$ has two singularities and one is the saddle point of $K$ and
another is a maximum point of $K$, $\omega$ has a zero of order $\alpha-1$,
$\alpha$ simple poles which are smooth extremal points of $K$ and a simple
pole which is the singularity of conical angle $2\pi\beta$. Suppose $K$ has
$I_{1}$ maximum points and $I_{2}+1$ minimum points, then
$I_{1}+I_{2}=\alpha$. Regard $S^{2}$ as $\mathbb{C}\cup\\{\infty\\}$. We can
assume that $0$ is the zero of $\omega$, $a_{1},\ldots,a_{I_{1}}$ are poles of
$\omega$ at which residues are negative and
$b_{1},\ldots,b_{I_{2}},b_{I_{2}+1}$ are poles of $\omega$ at which residues
are positive. Then
$Res_{a_{k}}(\omega)=\sigma,k=1,\ldots,I_{1},Res_{b_{I_{2}+1}}(\omega)=\sigma\lambda\beta~{}\text{and}~{}Res_{b_{l}}(\omega)=\sigma\lambda,l=1,\ldots,I_{2}$.
From these, we derive
$I_{1}\sigma+\sigma\lambda\beta+I_{2}\sigma\lambda=\sigma(I_{1}+\lambda\beta+I_{2}\lambda)=0$,
which implies $\lambda=-\frac{I_{1}}{I_{2}+\beta}$. Since
$\lambda=-\frac{2K_{1}+K_{2}}{2K_{2}+K_{1}}=-\frac{I_{1}}{I_{2}+\beta}<-1$, we
obtain $I_{1}>I_{2}+\beta$. Thus there is a nonzero $\widetilde{B}$ such that

$\omega=\frac{\widetilde{B}z^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=\sigma(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{\beta
I_{1}}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz.$

######  Proposition 4.6.

Let $\alpha,\beta\geq 2,I_{1}\geq 1$ and $I_{2}\geq 0$ be integers such that
$I_{1}+I_{2}=\alpha$ and $I_{1}>I_{2}+\beta$. Then, there exists a meromorphic
1-form $\omega$ on $S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined as follows

$\omega=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{\beta
I_{1}}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$
(11)

where $B\in\mathbb{C}\setminus\\{0\\}$ is a constant and
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}},b_{I_{2}+1}\in\mathbb{C}\setminus\\{0,1\\}$
are distinct complex numbers, if and only if one of the following conditions
holds.
(1) $I_{2}=0$;
(2) $I_{2}\geq 1,(I_{2}+\beta)\nmid I_{1}$ and
$(I_{2}+\beta)I_{1}>(\alpha-1){\rm GCD}(I_{2}+\beta,I_{1})$.

###### Proof.

Sufficiency

If $I_{2}=0$, a direct calculation verifies the result.

For $I_{2}\geq 1,(I_{2}+\beta)\nmid I_{1}$ and
$(I_{2}+\beta)I_{1}>(\alpha-1){\rm GCD}(I_{2}+\beta,I_{1})$, set $m={\rm
GCD}(I_{2}+\beta,I_{1})$. Then, by Theorem 2.6, there exists a meromorphic
function $f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$
of degree $\frac{I_{1}(I_{2}+\beta)}{m}$ and with branch data

$\Lambda=\\{(\underbrace{\frac{I_{2}+\beta}{m},\ldots,\frac{I_{2}+\beta}{m}}_{I_{1}}),(\frac{I_{1}\beta}{m},\underbrace{\frac{I_{1}}{m},\ldots,\frac{I_{1}}{m}}_{I_{2}}),(\alpha,1,\ldots,1)\\}.$

Without loss of generality, we can express $f$ as

$f(z)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}+\beta}{m}}}{(z-b_{I_{2}+1})^{\frac{I_{1}\beta}{m}}\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{m}}}$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}+1}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $C\neq 0$ is a constant. Then, a direct
calculation show that

$\frac{m}{I_{2}+\beta}\cdot\frac{df}{f}=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{\beta
I_{1}}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz,$

where $B\neq 0$ is a constants, is the desired meromorphic 1-form.

Necessity

Suppose there exists a meromorphic 1-form on
$S^{2}=\mathbb{C}\cup\\{\infty\\}$ defined by the form

$\omega=\frac{Bz^{\alpha-1}}{\prod_{k=1}^{I_{1}}(z-a_{k})\prod_{l=1}^{I_{2}+1}(z-b_{l})}dz=(\sum_{k=1}^{I_{1}}\frac{1}{z-a_{k}}-\frac{\frac{\beta
I_{1}}{I_{2}+\beta}}{z-b_{I_{2}+1}}-\frac{I_{1}}{I_{2}+\beta}\sum_{l=1}^{I_{2}}\frac{1}{z-b_{l}})dz$

where
$a_{1},\ldots,a_{I_{1}},b_{1},\ldots,b_{I_{2}+1}\in\mathbb{C}\setminus\\{0\\}$
are distinct complex numbers and $B\in\mathbb{C}\setminus\\{0\\}$ is a
constant.

If $I_{2}=0$, there is nothing to prove. If $I_{2}\geq 1$ and
$(I_{2}+\beta)\mid I_{1}$, then

$f(z)=exp(\int\omega)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})}{(z-b_{I_{2}+1})^{\frac{I_{1}\beta}{I_{2}+\beta}}\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{I_{2}+\beta}}}$

is a meromorphic function of degree $I_{2}$ on $\mathbb{C}\cup\\{\infty\\}$,
where $C\neq 0$ is a constant. Since $\omega=\frac{df}{f}$, we obtain
$I_{1}>\alpha-1$. However, $I_{1}+I_{2}=\alpha$. It is a contradiction.

Suppose $I_{2}\geq 1$ and $(I_{2}+\beta)\nmid I_{1}$. Set $m={\rm
GCD}(I_{2}+\beta,I_{1})$, then

$f(z)=exp(\int\frac{I_{2}+\beta}{m}\omega)=\frac{C\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}+\beta}{m}}}{(z-b_{I_{2}+1})^{\frac{I_{1}\beta}{m}}\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{m}}}$

is a meromorphic function of degree $\frac{I_{1}(I_{2}+\beta)}{m}$ on
$\mathbb{C}\cup\\{\infty\\}$, where $C\neq 0$ is a constant. The derivative of
$f$ is

$f^{\prime}(z)=\frac{CBz^{\alpha-1}\prod_{k=1}^{I_{1}}(z-a_{k})^{\frac{I_{2}+\beta}{m}-1}}{(z-b_{I_{2}+1})^{\frac{I_{1}\beta}{m}+1}\prod_{l=1}^{I_{2}}(z-b_{l})^{\frac{I_{1}}{m}+1}}.$

Then $\frac{(I_{2}+\beta)I_{1}}{m}>(\alpha-1)$, i.e.,
$(I_{2}+\beta)I_{1}>(\alpha-1){\rm GCD}(I_{2}+\beta,I_{1})$. ∎

By Proposition 4.6 and Theorem 2.2, there exists a non-CSC HCMU metric $g$ on
$S^{2}_{\\{\alpha,\beta\\}}$ such that the singularity of singular angle
$2\pi\alpha$ is the saddle point of the Gaussian curvature $K$ and the
singularity of singular angle $2\pi\beta$ is a minimum point of $K$.

## Acknowledgments

We would like to express our gratitude to the reviewers for their constructive
comments and suggestions, which have helped improve the quality and clarity of
this manuscript. Wei expresses his deep gratitude to Professor Yingyi Wu and
Professor Bin Xu for their invaluable encouragement and the insightful
discussions that greatly contributed to this work. This work is supported by
the National Natural Science Foundation of Henan (Grant No. 202300410067) and
partially supported by the National Natural Science Foundation of China (No.
12171140).

Declarations

Data Availability Statement This manuscript has no associated data.

Competing interests On behalf of all authors, the corresponding author states
that there is no conflict of interest.

## References

  * [1] E. Calabi, Extremal Kähler metrics. In: Seminar on Differential Geometry, Ann. of Math. Stud. 102, pp.259-290. Princeton Univ. Press, Princeton, 259-290 (1982).
  * [2] G. Boccara, Cycles comme produit de deux permutations de classes données, Discrete Math. 58 129-142 (1982)
  * [3] X.X. Chen, Weak limits of Riemannian metrics in surfaces with integral curvature bound, Calc. Var. Partial Differential Equations 6, 189-226 (1998).
  * [4] , Extremal hermitian metrics on Riemann surfaces, Calc. Var. Partial Differential Equations 8(3), 191-232 (1999).
  * [5] , Obstruction to the existence of metric whose curvature has umbilical Hessian in a K-Surface, Comm. Anal. Geom. 8(2), 267-299 (2000).
  * [6] Q. Chen, X.X. Chen and Y.Y. Wu, The structure of HCMU metric in a K-Surface, Int. Math. Res. Not. 2005(16), 941-958 (2005).
  * [7] Q. Chen, W. Wang, Y. Wu and B. Xu, Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces, Pac. J. Math. 273(1), 75-100 (2015).
  * [8] Q. Chen and Y.Y. Wu, Existences and explicit constructions of HCMU metrics on $S^{2}$ and $T^{2}$, Pac. J. Math. 240(2), 267-288 (2009).
  * [9] , Character 1-form and the existence of an HCMU metric, Math. Ann. 351(2), 327-345 (2011).
  * [10] Q. Chen, Y.Y. Wu and B. Xu, On one-dimensional and singular calabi’s extremal metrics whose Gauss curvatures have nonzero umblical Hessians, Isr. J. Math, 208, 385-412 (2015).
  * [11] A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Pro. Amer. Math. Soc. 131(11),3349-3355 (2004).
  * [12] , Metrics of constant positive curvature with four conic singularities on the sphere, Proc. Amer. Math. Soc. 148(9), 3957-3965 (2020).
  * [13] R. McOwen, Point sinularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103, 222-224 (1988).
  * [14] G. Mondello, D. Panov, Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math. Res. Not. (16), 4937-4995 (2016).
  * [15] L.G. Li, J.J. Song and B. Xu, Irreducible cone sphereical metrics and stable extensions of two line bundles, Adv. in Math. 388 (2021).
  * [16] C.S. Lin and X.H. Zhu, Explicit construction of extremal hermitian metrics with finite conical singularities on $S^{2}$, Comm. Anal. Geom. 10(1), 177-216 (2002).
  * [17] F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116, 1119-1129 (1992).
  * [18] J.J. Song, and B. Xu, On rational functions with more than three branch points, Algrbra Colloquium, 27(2), 231-246 (2020).
  * [19] M. Troyanov, Prescrbing curvature on compact surface with conical singularities, Tran. Amer. Math. Soc. 324(2), 793-821 (1991).
  * [20] M. Umehara and K. Yamada, Metric of constant curvature 1 with three conical singularities on the 2-sphere, Illinois J. Math. 44, 72-94 (2000).
  * [21] G.F. Wang and X.H. Zhu, Extremal hermitian metrics on Riemann surfaces with singularities, Duke Math. J, 104, 181-210 (2000).
  * [22] Z.Q. Wei and Y.Y. Wu, An existence theorem and energy intergral formula of HCMU metrics, Journal of University of Chinese Academy of Sciences, 33 (1), 16-22 (2016).
  * [23] , One existence theorem for non-CSC extremal Kähler metrics with singularities on $S^{2}$, Taiwan. J. Math, 22 (1), 55-62 (2018).
  * [24] , Non-CSC extremal Kähler metrics on $S^{2}$, Results Math, 74:58 (2019).
  * [25] G. Springer, Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., Addison-Wesley, 1957.

Yingjie Meng
School of Mathematics and Statistics
Henan University
Kaifeng 475004 P.R. China
<EMAIL_ADDRESS>

Zhiqiang Wei
School of Mathematics and Statistics
Henan University
Kaifeng 475004 P.R. China
Center for Applied Mathematics of Henan Province
Henan University
Zhengzhou 450046 P.R. China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>