It is a well-known consequence of the classical uniformization theorem that there is a metric with constant Gaussian curvature in each conformal class of any compact Riemann surface. It is natural to ask how to generalize this classical uniformization theory to compact surfaces with conical singularities and with nonempty boundary, or to find a "best metric" on such surfaces.

Instead of using metrics of constant curvature, Chen^[1-2] started to use the extremal Hermitian metrics to generalize the classical uniformization theory to Riemann surfaces with finite conical singularities. On any football there is at least an extremal Hermitian metric and it was claimed that there is at least an extremal Hermitian metric on any surface with boundary (see Ref. [2]).

Wang and Zhu^[3] discussed extremal Hermitian metrics with finite energy and area on Riemann surfaces with conical singularities and provided a classification of K_g for M with all singularities satisfying small enough angles. The primary challenge in obtaining this classification arises from the fact that the Gaussian curvature K_g may become unbounded near singular points. This challenge was overcome by studying the asymptotic behavior of the metrics near the singular points, along with conducting numerous meticulous analyzes of the metrics.

During above analytic process, Proposition 2.1 in Ref.[3], an accurate estimate of the conical structure of metrics, is necessary to obtain the asymptotic behavior of the Gaussian curvature and the estimate of the second covariant derivative of curvature. While showing that the Gaussian curvature can be extended continuously to singular points with small enough angles (specifically 2πα_i, provided that α_i≤1 for any i), Wang and Zhu^[3] combined the asymptotic behavior of the first derivatives of curvature and the holomorphicity of the second covariant derivative of curvature, and the proposition played an important role as a prerequisite. Also, without studying the conical structure of metrics, the approach to establish the estimate of the second covariant derivative of curvature will be invalid.

However, the proof of Proposition 2.1 given by Wang and Zhu remains several gaps that are eventually correct but lack essential explanation, which may create some obstacles for readers to understand. The main purpose of this paper is to make amends for the neglect and to provide a specific computation to verify such an estimate.

We now state Theorem A, i.e. Proposition 2.1 in Ref.[3], and Theorem B, a weaker estimate given by Chen^[2] that will be used to support the proof of Theorem A.

Let M be a compact Riemann surface with nonempty boundary ∂M. For any Hermitian metric g₀ on M, consider the set $\mathcal{G}$ (M) of metrics with the same area that are pointwise conformal to g₀ and agree with g₀ in a small neighborhood of ∂M. In the closure of this set $\mathcal{G}$(M) under some suitable H^{2, 2}-norm, we define the energy functional

$ E(g) = \int\limits_M K_g^2 \mathrm{d}g, $

where K_g is the Gaussian curvature of g and dg stands for the volume element. A critical point of this function is called an extremal Hermitian metric.

The Euler-Lagrange equation of this function is

$ \Delta_g K_g+K_g^2=C, $

(1)

for some constant C. Equivalently, in a local complex coordinate chart,

$ \frac{\partial}{\partial \bar{z}} K_{g, z z}=0, $

(2)

where K_{g, zz} is the second-order covariant derivative of K_g. The equations (1) and (2) are equivalent (see Ref.[1]). In particular, a metric with curvature satisfying K_{g, zz}=0 is called HCMU (this means that the Hessian of the Curvature of the Metric is Umbilical). There are examples of HCMU metrics that are not metrics of constant curvature in Ref.[1] and Wu^[5] studied the character 1-form of an HCMU metric.

Let D be a disk centered at the origin. Suppose that $g=\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ is an extremal metric with finite energy and area on $D-\{0\}$. Let $K$ be the Gaussian curvature of $g$. Then $\psi$ and $K$ satisfy the following system on $D-\{0\}$

$ \left\{\begin{array}{l} \Delta K=-K^2 \mathrm{e}^{2 \psi}+C \mathrm{e}^{2 \psi} , \\ \Delta \psi=-K \mathrm{e}^{2 \psi} . \end{array}\right. $

(3)

for some constant C.

Wang and Zhu considered the conical structure of extremal Hermitian metrics.

Theorem A (Proposition 2.1^[3]). Let $g= \mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ be an extremal metric in a punctured disk $D-\{0\}$ with finite area and energy and let $K$ be the Gaussian curvature of $g$. Then

$ \begin{aligned} \lim _{x \rightarrow 0}|x| K(x) \mathrm{e}^{\psi(x)}=0 . \end{aligned} $

The above estimate is an improved result of the following estimate given by Chen.

Theorem B (Theorem 2^[2]). Let $g=\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ be an extremal metric in a punctured disk $D-\{0\}$ with finite area and energy and let $K$ be the Gaussian curvature of $g$. Then

$ \begin{aligned} \lim _{x \rightarrow 0}|x|^{2} K(x) \mathrm{e}^{2 \psi(x)}=0. \end{aligned} $

1 Analysis results

To begin with, a technical method frequently appears in this paper and it is generalized as the following proposition, which shows an equivalent statement involving certain uniformity to continuity.

Proposition 1.1 For a given point $x_{0} \in \mathbb{R}^{n}$, suppose $f$ is defined on a neighborhood of $x_{0}$. Suppose $f$ is also defined on a bounded subset $\mathit{Ω}$ which satisfies:

1) $\operatorname{dist}\left(x_{0}, \mathit{Ω}\right)>0$,

2) $\mathit{Ω}$ is nonempty along every line through $x_{0}$, i. e. $\forall y \neq 0$,

$ \mathit{Ω} \cap\left\{x_{0}+t y \mid t \in \mathbb{R}\right\} \neq \varnothing . $

Then f is continuous at x₀ if and only if for an arbitrary sequence of real numbers r_i→0,

$ \begin{aligned} \lim _{i \rightarrow \infty} f\left(x_{0}+r_{i} x\right)=f\left(x_{0}\right), \end{aligned} $

uniformly for $x \in \mathit{Ω}$.

Proof Without loss of generality, we can set x₀=0.

We first prove the necessity. Denote $d= \sup _{x \in \mathit{Ω}}|x|>0$. For any $\varepsilon>0$, since $f$ is continuous at 0, there exists $\delta>0$ such that if $|x|<\delta$, then

$ |f(x)-f(0)|<\varepsilon . $

Taking an arbitrary sequence$\left\{r_{i}\right\} \rightarrow 0$, one knows that there exists an integer $N>0$ such that for any $i>N$

$ \left|r_{i}\right|<\frac{\delta}{d} . $

Therefore, for all $x \in \mathit{Ω}$, if $i>N$, then

$ \left|f\left(r_{i} x\right)-f(0)\right|<\varepsilon, $

which completes one direction of the proposition.

For sufficiency, since $\operatorname{dist}(0, \mathit{Ω})>0$, we can suppose $\operatorname{dist}(0, \mathit{Ω})>\varepsilon>0$. Taking an arbitrary sequence $\left\{x_{i}\right\} \rightarrow 0$, there exists a real number $\left|\lambda_{i}\right|>\varepsilon$ for each $i$ such that

$ \lambda_{i} \frac{x_{i}}{\left|x_{i}\right|} \in \mathit{Ω} . $

The existence is guaranteed by the property 2) of $\mathit{Ω}$. Then we denote $r_{i}=\frac{\left|x_{i}\right|}{\lambda_{i}}$ satisfying

$ \left|r_{i}\right|<\frac{\left|x_{i}\right|}{\varepsilon} \rightarrow 0 \quad \text { as } i \rightarrow \infty . $

Hence, by the uniformity, as $i \rightarrow \infty$,

$ \begin{aligned} \left|f\left(x_{i}\right)-f(0)\right| & =\left|f\left(r_{i} \cdot \lambda_{i} \frac{x_{i}}{\left|x_{i}\right|}\right)-f(0)\right| \\ & \leqslant \sup _{x \in \mathit{Ω}}\left|f\left(r_{i} x\right)-f(0)\right| \rightarrow 0 . \end{aligned} $

For the arbitrariness of $\left\{x_{i}\right\}, f$ is continuous at 0.

Remark 1.1 From the proof, the necessity holds for more general Ω without property 2).

Most uniformity in this paper is from Proposition 1.1, so readers should keep this in mind when encountering uniformity in the sequel.

Next, we will introduce the Newtonian potential (also called Poisson potential) and show some basic properties of it in the context of $ \mathbb{R}^2 $. To do so, we need the following theorem known as Young's inequality for convolution, which offers us a tool to estimate the convolution.

Theorem 1.1 (Corollary 2.25^[6]). Let $p^{\prime}$, $q^{\prime}, r^{\prime} \geqslant 1$ and suppose that $\left(1 / p^{\prime}\right)+\left(1 / q^{\prime}\right)=1+ \left(1 / r^{\prime}\right)$. If $u \in L^{p^{\prime}}\left(\mathbb{R}^{n}\right)$ and $v \in L^{q^{\prime}}\left(\mathbb{R}^{n}\right)$, then $u * v \in L^{r^{\prime}}\left(\mathbb{R}^{n}\right)$, and

$ \|u * v\|_{r^{\prime}} \leqslant\|u\|_{p^{\prime}}\|v\|_{q^{\prime}}. $

Denote

$ \mathit{\Gamma}(x)=\frac{1}{2 \pi} \ln |x| \quad \text { on } \mathbb{R}^{2}-\{0\}, $

known as the fundamental solution of the Laplacian equation and in the sequel. For an integrable function f, the integral

$ \mathit{\Gamma} * f(x)=\int_{\mathbb{R}^{2}} \mathit{\Gamma}(x-y) f(y) \mathrm{d} y $

is called the Newtonian potential of f. Then we give the following proposition which claims that the Laplacian (in the weak sense) of the Newtonian potential is equal to the function f itself almost everywhere.

Proposition 1.2 Suppose that Ω is a bounded domain of $\mathbb{R}^{2}$ and $f \in L^{1}(\mathit{Ω})$.Then $\mathit{\Gamma} * f$ is well－defined as an integrable function, and the following equality holds: for any test function $\phi \in C_{0}^{\infty}(\mathit{Ω})$,

$ \int_{\mathit{Ω}}(\mathit{\Gamma} * f) \cdot \Delta \phi \mathrm{d} x=\int_{\mathit{Ω}} f \phi \mathrm{~d} x. $

Proof First, we prove that $\mathit{\Gamma} * f \in L^{1}(\mathit{Ω})$. Let $B_{r}$ be a disk at the origin with radius $r>0$.Take $r>\operatorname{diam}(\mathit{Ω})$ for $B_{r}$.Since $\mathit{\Gamma} \in L_{\text {loc }}^{1}\left(\mathbb{R}^{2}\right)$, denote $G =\int_{B_{r}} \mathit{\Gamma}(x) \mathrm{d} x<\infty$. Then by Fubini's theorem, we have

$ \begin{aligned} \int_{\mathit{Ω}}|\mathit{\Gamma} * f| \mathrm{d} x & \leqslant \int_{\mathit{Ω}}|f(y)| \int_{\mathit{Ω}}|\mathit{\Gamma}(x-y)| \mathrm{d} x \mathrm{~d} y \\ & \leqslant G \int_{\mathit{Ω}}|f(y)| \mathrm{d} y=G\|f\|_{L^{1}(\mathit{Ω})} <\infty. \end{aligned} $

Thus, Γ*f is well-defined as an integrable function.

To calculate the Laplacian of f, we use the approximation of f (see Corollary 2.30^[6]). Specifically, a sequence $\left\{f_{i}\right\}$ in $C_{0}^{\infty}(\mathit{Ω})$ satisfies that as $i \longrightarrow \infty, f_{i} \rightarrow f$ in $L^{1}(\mathit{Ω})$.Denote $\chi_{r}$ the characteristic function of $B_{r}$.For each $i$, we have $\left(\chi_{r} \mathit{\Gamma}\right) * f_{i}=\mathit{\Gamma} * f_{i} \quad$ and $\quad\left(\chi_{r} \mathit{\Gamma}\right) * f=\mathit{\Gamma} * f \quad$ on $\mathit{Ω}$. The zero extensions of $f_{i}, f$ on $\mathbb{R}^{2}$ are given by assigning $f_{i}=0, f=0$ on $\mathbb{R}^{2}-\mathit{Ω}$, i.e.

$ f_{i}= \begin{cases}f_{i} & \text { on } \mathit{Ω}, \\ 0 & \text { on } \mathbb{R}^{2}-\mathit{Ω},\end{cases} $

and

$ f= \begin{cases}f & \text { on } \mathit{Ω}, \\ 0 & \text { on } \mathbb{R}^{2}-\mathit{Ω}.\end{cases} $

We shall still denote the above extensions by $f_{i}, f$. Then by Theorem 1.1 with $p^{\prime}=q^{\prime}=r^{\prime}=1$, we know that as $i \rightarrow \infty$,

$ \begin{aligned} \left\|\mathit{\Gamma} * f_{i}-\mathit{\Gamma} * f\right\|_{L^{1}(\mathit{Ω})} & \leqslant\left\|\chi_{r} \mathit{\Gamma}\right\|_{L^{1}\left(\mathbb{R}^{2}\right)} \cdot\left\|f_{i}-f\right\|_{L^{1}\left(\mathbb{R}^{2}\right)} \\ & =\|\mathit{\Gamma}\|_{L^{1}\left(B_{r}\right)} \cdot\left\|f_{i}-f\right\|_{L^{1}(\mathit{Ω})} \rightarrow 0. \end{aligned} $

Thus, $\mathit{\Gamma} * f_{i} \rightarrow \mathit{\Gamma} * f$ in $L^{1}(\mathit{Ω})$. In particular, $f_{i}$ and $\mathit{\Gamma} * f_{i}$ weakly converge respectively to $f$ and $\mathit{\Gamma} * f$, i.e.for any $g \in L^{\infty}(\mathit{Ω})$,

$ \begin{aligned} \lim _{i \rightarrow \infty} \int_{\mathit{Ω}} f_{i} g \mathrm{~d} x=\int_{\mathit{Ω}} f g \mathrm{~d} x \end{aligned} $

and

$ \begin{aligned} \lim _{i \rightarrow \infty} \int_{\mathit{Ω}}\left(\mathit{\Gamma} * f_{i}\right) \cdot g \mathrm{~d} x=\int_{\mathit{Ω}}(\mathit{\Gamma} * f) \cdot g \mathrm{~d} x. \end{aligned} $

On the other hand, since $f_{i} \in C_{0}^{\infty}(\mathit{Ω})$ for each $i$, by Theorem 1 ^[7], we have

$ \begin{aligned} \Delta\left(\mathit{\Gamma} * f_{i}\right)=f_{i} . \end{aligned} $

Therefore, for any test function $\phi \in C_{0}^{\infty}(\mathit{Ω})$, we have

$ \int_{\mathit{Ω}}\left(\mathit{\Gamma} * f_{i}\right) \cdot \Delta \phi \mathrm{d} x=\int_{\mathit{Ω}} f_{i} \phi \mathrm{~d} x. $

Thus, for any test function $\phi \in C_{0}^{\infty}(\mathit{Ω})$,

$ \begin{aligned} \int_{\mathit{Ω}}(\mathit{\Gamma} * f) \cdot \Delta \phi \mathrm{d} x & =\lim _{i \rightarrow \infty} \int_{\mathit{Ω}}\left(\mathit{\Gamma} * f_{i}\right) \cdot \Delta \phi \mathrm{d} x \\ & =\lim _{i \rightarrow \infty} \int_{\mathit{Ω}} f_{i} \cdot \phi \mathrm{~d} x \\ & =\int_{\mathit{Ω}} f \phi \mathrm{~d} x. \end{aligned} $

Corollary 1.1 Let D be a disk at the origin and $D^{*}=D-\{0\}$. Suppose $f \in C^{\infty}\left(D^{*}\right) \cap L^{1}(D)$. Then we have that $\mathit{\Gamma} * f$ is smooth on $D^{*}$ and

$ \Delta(\mathit{\Gamma} * f)=f \quad \text { on } \quad D^{*}. $

Proof Take r>diam(D) for B_r. Denoting the zero extensions of f on $\mathbb{R}^{2}$ by $f$, consider the L²-norm of f. Similar to the proof of Proposition 1.2, by Theorem 1.1, we have

$ \|\mathit{\Gamma} * f\|_{L^{2}(D)} \leqslant\|\mathit{\Gamma}\|_{L^{2}\left(B_{r}\right)}\|f\|_{L^{1}(D)}<\infty . $

Thus, $\mathit{\Gamma} * f \in L^{2}\left(D^{*}\right)$.By Proposition 1.2, for any $\phi \in C_{0}^{\infty}\left(D^{*}\right)$，

$ \int_{D}(\mathit{\Gamma} * f) \cdot \Delta \phi \mathrm{d} x=\int_{D} f \phi \mathrm{~d} x. $

Then by Weyl's Lemma (see Proposition 28^[8]), we have: $\mathit{\Gamma} * f \in C^{\infty}\left(D^{*}\right)$ and

$ \Delta(\mathit{\Gamma} * f)=f \quad \text { on } \quad D^{*} . $

One of the gaps in Chen's previous proof for Theorem B is that they lacked enough explanation on how to restrain the L^p-norm of a function by the L¹-norm of its Laplacian. The next proposition given by us will provide concrete evidence. To prove the proposition, we list some related theorems.

The following result proved by Gilbarg and Trudinger embraces a special case of the Calderon-Zygmund inequality.

Theorem 1.2 (Theorem 9.9^[9]). Let Ω be a bounded domain in $\mathbb{R}^{n}$. Let $f \in L^p(\mathit{Ω}), 1 <p <\infty$, and let w be the Newtonian potential of f. Then $w \in W^{2, p}(\mathit{Ω}), \Delta w=f \text { a. e. }$ and

$ \begin{aligned} \left\|D^2 w\right\|_p \leqslant C\|f\|_p, \end{aligned} $

where C depends only on n and p.

Notice that p above is required to be larger than 1, our proposition 1.2 indicates that the above inequality may not hold but the equality of f and the Laplacian of its Newtonian potential remains true as for p=1.

Theorem 1.3 (Theorem 9.11^[9]). Let Ω be an open set in $\mathbb{R}^{n}$ and $u \in W_{\text {loc }}^{2, p}(\mathit{Ω}) \cap L^{p}(\mathit{Ω}), 1< p<\infty$, a strong solution of the equation $L u=f$ in $\mathit{Ω}$ where the coefficients of $L$ satisfy, for positive constants $\lambda, \Lambda$,

$ \begin{aligned} & a^{i j} \in C^{0}(\mathit{Ω}), \quad b^{i}, c \in L^{\infty}(\mathit{Ω}), \quad f \in L^{p}(\mathit{Ω}) ; \\ & a^{i j} \xi_{i} \xi_{j} \geqslant \lambda|\xi|^{2} \quad \forall \xi \in \mathbb{R}^{n} ; \\ & \left|a^{i j}\right|, \left|b^{i}\right|, |c| \leqslant \Lambda, \end{aligned} $

where $i, j=1, \cdots, n$. Then for any domain $\mathit{Ω}^{\prime} \subset \subset \mathit{Ω}$,

$ \|u\|_{2, p ; \mathit{Ω}^{\prime}} \leqslant C\left(\|u\|_{p ; \mathit{Ω}}+\|f\|_{p ; \mathit{Ω}}\right), $

where $C$ depends on $n, p, \lambda, \Lambda, \mathit{Ω}^{\prime}, \mathit{Ω}$ and the moduli of continuity of the coefficients $a^{i j}$ on $\mathit{Ω}^{\prime}$.

This gives interior estimates for strong solutions (not classical solutions) of second-order elliptic equations. For more details, see Chapter 9 in Ref.[9].

The proof of our proposition also requires the Sobolev embedding theorem. To state the embedding theorem, we suppose that Ω is a domain in $\mathbb{R}^{n}$ and introduce the cone condition.

Definition 1.1 A domain Ω satisfies the cone condition if there exists a finite cone C such that each x∈Ω is the vertex of a finite cone C_x contained in Ω and congruent to C.

For a domain $\mathit{Ω}$, we define $C_{B}^{m}(\mathit{Ω})$ to consist of those functions $\phi \in C^{m}(\mathit{Ω})$ for which $D^{\alpha} u$ is bounded on $\mathit{Ω}$ for $0 \leqslant|\alpha| \leqslant m . C_{B}^{m}(\mathit{Ω})$ is a Banach space with norm given by

$ \begin{aligned} \|\phi\|_{C_{B}^{m}(\mathit{Ω})}=\max _{0 \leqslant \alpha \leqslant m} \sup _{x \in \mathit{Ω}}\left|D^{\alpha} \phi(x)\right| . \end{aligned} $

The following theorem is known as the Sobolev embedding theorem.

Theorem 1.4 (Theorem 4.12^[6]). Let Ω be a domain in $\mathbb{R}^{n}$. Let $j \geqslant 0$ and $m \geqslant 1$ be integers and let $1 \leqslant p<\infty$. Suppose $\mathit{Ω}$ satisfies the cone condition. If either $m p>n$ or $m=n$ and $p=1$, then

$ W^{j+m, p}(\mathit{Ω}) \hookrightarrow C_{B}^{j}(\mathit{Ω}) . $

Moreover,

$ W^{m, p}(\mathit{Ω}) \hookrightarrow L^{q}(\mathit{Ω}) \quad \text { for } p \leqslant q \leqslant \infty . $

For Sobolev space $W^{m, p}(\mathit{Ω})$ with $p=2$, we denote it by $H^{m}(\mathit{Ω})$.

Now we are ready to state our another analytic result required to prove Theorem A and present the proof.

Proposition 1.3 Let Ω be a bounded open subset of $\mathbb{R}^2$ with sufficiently smooth boundary and p>2 be a real number. If a sequence of functions$\left\{F_{i}\right\} \subset W^{2, p}(\mathit{Ω})$, satisfies

$ \left\|F_{i}\right\|_{L^{2}(\mathit{Ω})} \rightarrow 0 \quad \text { and } \quad\left\|\Delta F_{i}\right\|_{L^{1}(\mathit{Ω})} \rightarrow 0 , $

as $i \rightarrow \infty$, then for any domain $\mathit{Ω}^{\prime} \subset \subset \mathit{Ω}$ satisfying the cone condition, we have

$ \left\|F_{i}\right\|_{L^{p}\left(\mathit{Ω}^{\prime}\right)} \rightarrow 0 \quad \text { as } i \rightarrow \infty . $

Proof For each $i \in \mathbb{N}$, denote $f_{i}=\Delta F_{i} \in L^{p}(\mathit{Ω})$ with taking weak derivatives. In order to prove Proposition 1.3, we split F_i into two parts

$ F_{i}=u_{i}+v_{i} \quad \text { on } \quad \mathit{Ω}, $

where $u_{i}$ is the Newtonian potential of $f_{i}$. Denote the zero extension of $f_{i}$ by $\widetilde{f_{i}}$, given by assigning $\widetilde{f_{i}}=0$ on $\mathbb{R}^{2}-\mathit{Ω}$. Taking $r>\operatorname{diam} \mathit{Ω}$, then the modified Newtonian potential is given by

$ \widetilde{u_{i}}(x)=\left(\chi_{r} \mathit{\Gamma}\right) * \widetilde{f_{i}}(x), \quad x \in \mathbb{R}^{2}. $

Applying Theorem 1.1, the Young's convolution inequality, with $r^{\prime}=p^{\prime}=p, q^{\prime}=1$, there exists a constant $c_{1}$ (independent of $i$) such that $\forall i \in \mathbb{N}$,

$ \left\|{\widetilde{u_{i}}}\right\|_{L^{p}(\mathit{Ω})} \leqslant\|\mathit{\Gamma}\|_{L^{p}\left(B_{r}\right)}\left\|f_{i}\right\|_{L^{1}(\mathit{Ω})} \leqslant c_{1}\left\|f_{i}\right\|_{L^{1}(\mathit{Ω})}. $

Taking $i \rightarrow \infty$, we have

$ \left\|\widetilde{u_{i}}\right\|_{L^{p}(\mathit{Ω})} \leqslant c_{1}\left\|f_{i}\right\|_{L^{1}(\mathit{Ω})}=c_{1}\left\|\Delta F_{i}\right\|_{L^{1}(\mathit{Ω})} \rightarrow 0. $

On the other hand, denote $u_{i}=\mathit{\Gamma} * f_{i}$ and then by the Theorem 1.2 given by Gilbarg and Trudinger^[9], one has $u_{i} \in W^{2, p}(\mathit{Ω})$ for each i and

$ \Delta u_{i}=f_{i} . $

Due to the definition of $\widetilde{u_i}$ and u_i, we have

$ \widetilde{u_{i}}=u_{i} \quad \text { on } \quad \mathit{Ω} . $

Thus, $v_{i} \in W^{2, p}$ and $\Delta v_{i}=0$.As $i \rightarrow \infty$, one has

$ \begin{aligned} \left\|v_{i}\right\|_{L^{2}(\mathit{Ω})} & =\left\|F_{i}-u_{i}\right\|_{L^{2}(\mathit{Ω})} \\ & \leqslant\left\|F_{i}\right\|_{L^{2}(\mathit{Ω})}+\left\|u_{i}\right\|_{L^{2}(\mathit{Ω})} \rightarrow 0 . \end{aligned} $

By Theorem 1.3, where p=2, the operator $L= \Delta$ and $f=0$, we have for any $\mathit{Ω}^{\prime}$ as a compact sub-domain of Ω satisfying the cone condition, there exists a constant c₂ (independent of i) such that

$ \left\|v_{i}\right\|_{H^{2}\left(\mathit{Ω}^{\prime}\right)} \leqslant c_{2}\left\|v_{i}\right\|_{L^{2}(\mathit{Ω})} . $

Therefore, as $i \rightarrow \infty$,

$ \left\|v_{i}\right\|_{H^{2}\left(\mathit{Ω}^{\prime}\right)} \rightarrow 0 . $

Furthermore, by Theorem 1.4, the Sobolev embedding theorem, we have $H^{2}\left(\mathit{Ω}^{\prime}\right) \leftrightarrow L^{p}\left(\mathit{Ω}^{\prime}\right)$.It follows that $\left\|v_{i}\right\|_{L^{p}\left(\mathit{Ω}^{\prime}\right)} \rightarrow 0$ as $i \rightarrow \infty$. Thus,

$ \left\|F_{i}\right\|_{L^{p}\left(\mathit{Ω}^{\prime}\right)} \leqslant\left\|u_{i}\right\|_{L^{p}\left(\mathit{Ω}^{\prime}\right)}+\left\|v_{i}\right\|_{L^{p}\left(\mathit{Ω}^{\prime}\right)} \rightarrow 0 . $

2 Proof of Theorem A and Theorem B

Now we will show the complete proofs of Theorem A and Theorem B and explain how we apply our results to the proofs of Theorem A and B.

First, we introduce some notations. For any given $a>0$, let $T_{a}=\left\{x\left|\mathrm{e}^{-a}<|x|<\mathrm{e}^{a}\right\}\right. $. Let $D$ be a disk centered at the origin. Suppose that $g= \mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ is an extremal metric with finite energy and area on $D-\{0\}$. Let $\left\{x_{i}\right\}$ be a sequence on $D-\{0\}$ satisfying $\left\{x_{i}\right\} \rightarrow 0$, as $i \rightarrow \infty$. To study the asymptotic behavior of the metric, Wang and Zhu ^[3] considered a sequence of contracting domains $\left|x_{i}\right| T_{1}$. For convenience, they made $\left|x_{i}\right| T_{1}$ transformed into $T_{a}$ with the extremal metrics to maintain the form. Specifically, define associated sequences of metrics $\left\{\psi_{i}\right\}$ and functions $\left\{K_{i}\right\}$ on $T_{2}$ as

$ \begin{gathered} \psi_i(y)=\psi\left(\left|x_i\right| y\right)+\ln \left|x_i\right| \quad \text { and } \\ K_i(y)=K\left(\left|x_i\right| y\right) . \end{gathered} $

(4)

By equation (3), $K_{i}(x)$ is the curvature function of metric $\mathrm{e}^{2 \psi_{i}}|\mathrm{~d} z|^{2}$ in the domain $T_{2}$, i.e.

$ \Delta \psi_{i}=-K_{i} \mathrm{e}^{2 \psi_{i}} . $

Without other declaration, $\psi_{i}$ and $K_{i}$ will always stand the same with the above.

2.1 Proof of Theorem B

It is necessary to figure out the asymptotic behavior of metrics ψ_i, given by those coordinate transformations.

Lemma 2.1 (Theorem 2^[4]). Let $g= \mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ be an extremal metric in a punctured disk D-{0} with finite area and energy and ψ_i is defined as before. Then we have

$ \lim _{i \rightarrow \infty} \psi_{i}(y)=-\infty $

uniformly for $y \in T_{1}$.

The following lemma is an improved result of Chen's estimate (see Ref.[2]).

Lemma 2.2 (Lemma 1.1^[3]). Let $g= \mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ be an extremal metric with finite energy and area on D-{0}. Then

$ \begin{aligned} \lim _{r \rightarrow 0} \sup _{x, x^{\prime} \in r T_{1}} \frac{\psi(x)+\ln |x|}{\psi\left(x^{\prime}\right)+\ln \left|x^{\prime}\right|}=1 . \end{aligned} $

Although the proof of Theorem B was shown by Chen^[2], it omitted several important steps. Here we reprove this estimate and will fix all the gaps.

Proof of Theorem B For any sequence of points $\left\{x_{i}\right\} \rightarrow 0$, we want to show

$ \begin{aligned} \lim _{i \rightarrow \infty}\left|K\left(x_{i}\right)\right| \mathrm{e}^{2\left(\psi\left(x_{i}\right)+\ln \left|x_{i}\right|\right)}=0 . \end{aligned} $

For each $x_{i}$, define $\left\{\psi_{i}\right\}$ and $\left\{K_{i}\right\}$ in $T_{1}$ as equation(4).Then $K_{i}(y)=K\left(y\left|x_{i}\right|\right)$ satisfies the extremal equation

$ \Delta K_{i}(y)+K_{i}(y)^{2} \mathrm{e}^{2 \psi_{i}(y)}=C \mathrm{e}^{2 \psi_{i}(y)} . $$ $

Set

$ a_{i}=\inf _{y \in T_{1}} \psi_{i}(y) \quad \text { and } \quad \tilde{K}_{i}(y)=K_{i}(y) \mathrm{e}^{a_{i}} . $

Then

$ \left\|\widetilde{K}_{i}^{2}\right\|_{L^{2}\left(T_{1}\right)}=\int_{T_{1}} K_{i}^{2} \mathrm{e}^{2 a_{i}} \mathrm{~d} y \leqslant \int_{T_{1}} K_{i}^{2} \mathrm{e}^{2 \psi_{i}} \mathrm{~d} y . $

Since the original metric $\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ has finite energy on $D-\{0\}$, it follows that

$ \begin{aligned} \lim _{i \rightarrow \infty} \int_{T_{1}} K_{i}^{2} \mathrm{e}^{2 \psi_{i}} \mathrm{~d} y=\lim _{i \rightarrow \infty} \int_{\left|x_{i}\right| T_{1}} K^{2} \mathrm{e}^{2 \psi} \mathrm{~d} x=0 . \end{aligned} $

Thus as $i \rightarrow \infty$ we have

$ \left\|\tilde{K}_i\right\|_{L^2\left(T_1\right)} \rightarrow 0 . $

(5)

On the other hand,

$ \Delta \tilde{K}_{i}(y)+\tilde{K}_{i}(y) K_{i}(y) \mathrm{e}^{2 \psi_{i}(y)}=C \mathrm{e}^{a_{i}} \mathrm{e}^{2 \psi_{i}}, \quad i \in \mathbb{N}, y \in T_{1}. $

Denote $f_{i}(y)=-\widetilde{K}_{i}(y) K_{i}(y) \mathrm{e}^{2 \psi_{i}(y)}+C \mathrm{e}^{a_{i}} \mathrm{e}^{2 \psi_{i}(y)}$, i.e. $f_{i}=\Delta \tilde{K}_{i}$. By Lemma 2.1, there exists $b<0$ such that for sufficiently large $i$,

$ a_{i}<b. $

Then we have: for sufficiently large i,

$ \begin{aligned} \int_{T_{1}}\left|f_{i}\right| \mathrm{d} y & \leqslant \mathrm{e}^{b}\left(\int_{T_{1}} K_{i}^{2} \mathrm{e}^{2 \psi_{i}} \mathrm{~d} y+|C| \int_{T_{1}} \mathrm{e}^{2 \psi_{i}} \mathrm{~d} y\right) \\ & =\mathrm{e}^{b}\left(\int_{\left|x_{i}\right| T_{1}} K^{2} \mathrm{e}^{2 \psi} \mathrm{~d} x+|C| \int_{\left|x_{i}\right| T_{1}} \mathrm{e}^{2 \psi} \mathrm{~d} x\right). \end{aligned} $

Therefore, with the original metric $\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ having finite energy and area, as $i \rightarrow \infty$,

$ \left\|\Delta \tilde{K}_{i}\right\|_{L^{1}\left(T_{1}\right)}=\int_{T_{1}}\left|f_{i}\right| \mathrm{d} y \rightarrow 0 $

(6)

By the Hölder inequality, for any 1 < s < 2,

$ \left\|f_{i}\right\|_{L^{s}\left(T_{1 / 2}\right)} \leqslant\left\|\tilde{K}_{i}\right\|_{L^{\frac{2 s}{2-s}}\left(T_{1 / 2}\right)}\left\|K_{i} \mathrm{e}^{2 \psi_{i}}\right\|_{L^{2}\left(T_{1 / 2}\right)}+ \\ \left\|C \mathrm{e}^{a_{i}} \mathrm{e}^{2 \psi_{i}}\right\|_{L^{s}\left(T_{1 / 2}\right)} $

(7)

Denote $p=\frac{2 s}{2-s}$. Then we have $p>2$ since $1<s<$ 2; furthermore, since $\tilde{K}_{i} \in C^{\infty}\left(\bar{T}_{1}\right)$, then $\left\{\tilde{K}_{i}\right\} \subset W^{2, p}\left(T_{1}\right)$. Combined with limits (5) and (6), by Proposition 1.3: as $i \rightarrow \infty$,

$ \left\|\tilde{K}_{i}\right\|_{L^{p}\left(T_{1 / 2}\right)} \rightarrow 0 . $

(8)

For sufficiently large i, with the original metric having finite energy and area, we have the following limits

$ \begin{aligned} & \left\|K_{i} \mathrm{e}^{2 \psi_{i}}\right\|_{L^{2}\left(T_{1 / 2}\right)}^{2} \leqslant \mathrm{e}^{2 b} \int_{\left|x_{i}\right| T_{1 / 2}} K^{2} \mathrm{e}^{2 \psi} \mathrm{~d} x \rightarrow 0, \\ & \left\|C \mathrm{e}^{a_{i}} \mathrm{e}^{2 \psi_{i}}\right\|_{L^{s}\left(T_{1 / 2}\right)}^{S} \leqslant C \mathrm{e}^{4 b} \int_{T_{1 / 2}} \mathrm{e}^{2 \psi_{i}} \mathrm{~d} y \rightarrow 0. \end{aligned} $

Therefore, we have estimated all the terms in (7) and obtain that as $i \rightarrow \infty$,

$ \left\|\tilde{K}_{i}\right\|_{L^{s}\left(T_{1 / 2}\right)}=\left\|f_{i}\right\|_{L^{s}\left(T_{1 / 2}\right)} \rightarrow 0 . $

(9)

For 1 < s < 2, by Theorem 1.3 with $p=s, L= \Delta$ and $T_{1 / 4} \subset \subset T_{1 / 2}$, and combined (8) and (9), we have: there exists $c_{3}$ independent of $i$ such that

$ \left\|\tilde{K}_{i}\right\|_{W^{2, s}\left(T_{1 / 4}\right)} \leqslant c_{3}\left(\left\|\tilde{K}_{i}\right\|_{L^{s}\left(T_{1 / 4}\right)}+\left\|f_{i}\right\|_{L^{s}\left(T_{1 / 4}\right)}\right) \rightarrow 0 . $

In particular, by Theorem 1.4, the embedding theorems, $W^{2, s}\left(T_{1 / 4}\right) \leftrightarrow C_{B}^{0}\left(T_{1 / 4}\right)$, which implies

$ \begin{aligned} \sup _{y \in T_{1 / 4}}\left|\tilde{K}_i(y)\right| \rightarrow 0. \end{aligned} $

(10)

From Lemma 2.2, we have

$ \begin{aligned} & \lim _{i \rightarrow+\infty} \sup _{x, x^{\prime} \in\left|x_{i}\right| T_{1}}\left|\frac{\psi(x)+\ln |x|}{\psi\left(x^{\prime}\right)+\ln \left|x^{\prime}\right|}\right| \\ = & \lim _{i \rightarrow+\infty} \sup _{x, x^{\prime} \in T_{1}}\left|\frac{\psi\left(\left|x_{i}\right| x\right)+\ln \left|x_{i}\right|}{\psi\left(\left|x_{i}\right| x^{\prime}\right)+\ln \left|x_{i}\right|}\right| \\ = & \lim _{i \rightarrow+\infty} \sup _{x, x^{\prime} \in T_{1}}\left|\frac{\psi_{i}(x)}{\psi_{i}\left(x^{\prime}\right)}\right|=1 . \end{aligned} $

Thus,

$\frac{a_{i}}{\psi_{i}(y)} \leqslant \frac{\inf _{T_{1}} \psi_{i}}{\sup _{T_{1}} \psi_{i}}=\left(\sup _{x, x^{\prime} \in T_{1}}\left|\frac{\psi_{i}(x)}{\psi_{i}\left(x^{\prime}\right)}\right|\right)^{-1} \rightarrow 1 \quad$ as $i \rightarrow \infty$. Therefore, for sufficiently large $i$, we have

$ a_{i} \geqslant 2 \psi_{i}(y), \quad \forall y \in T_{1} . $

Let $y_{i}=\frac{x_{i}}{\left|x_{i}\right|}$, this implies that

$ \left|x_{i}\right|^{2}\left|K\left(x_{i}\right)\right| \mathrm{e}^{2 \psi\left(x_{i}\right)}=\left|K_{i}\left(y_{i}\right)\right| \mathrm{e}^{2 \psi_{i}\left(y_{i}\right)} \leqslant\left|K_{i}\left(y_{i}\right)\right| \mathrm{e}^{a_{i}}. $

Since $\left|K_{i}(y) \mathrm{e}^{a_{i}}\right| \leqslant \sup _{T_{1 / 4}}\left|\tilde{K}_{i}\right|, \forall y \in T_{1 / 4}$ and combine (10), we obtain that

$ \left|K_{i}\left(y_{i}\right)\right| \mathrm{e}^{a_{i}} \longrightarrow 0 \quad \text { as } i \rightarrow \infty . $

(11)

2.2 Proof of Theorem A

Chen^[4] studied the behavior of the Newtonian potential satisfying certain asymptotic properties near the origin.

Lemma 2.3 Let$D \subset R^{2}$ be a disk and let $f \in L^{1}(D)$. Suppose that $\begin{aligned}\lim _{|x| \rightarrow 0}|x|^{2} f(x)=0\end{aligned}$. Define $v(x)$ from the Poisson potential of $f$ by

$ v(x)=\int_{D} \mathit{\Gamma}(x-y) f(y) \mathrm{d} y . $

Then the following two statements hold:

1) $\begin{aligned}\lim _{|x| \rightarrow 0} \frac{v(x)}{\ln |x|}=0\end{aligned}$,

2) $\begin{aligned}\lim _{r \rightarrow 0|x|} \sup _{=\left|x^{\prime}\right|=r}\left(v(x)-v\left(x^{\prime}\right)\right)=0\end{aligned}$.

Define the oscillation of a function $f$ on a subset $U \subset \mathbb{R}^{n}$ by

$ \begin{aligned} \operatorname{Osc}_{x \in U}\{f(x)\}=\sup _{x \in U} f(x)-\inf _{x \in U} f(x) . \end{aligned} $

The following lemma is equivalent to Theorem 3^[3] and shows that the metric is asymptotically rotationally symmetric.

Lemma 2.4 Let $g=\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}$ be an extremal metric with finite energy and area on $D-\{0\}$ and $\psi_{i}$ is defined as before. Then under polar coordinates (the range of the θ will always be [0, 2π) if omitted),

$ \begin{aligned} \lim _{i \rightarrow \infty} \operatorname{Osc}_{\theta}\left\{\psi_{i}(r, \theta)\right\}=0, \end{aligned} $

uniformly for $\mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$

Proof First from (3) and the Hölder inequality, we notice that

$ \begin{aligned} \int_{D-\{0\}}|\Delta \psi| \mathrm{d} x & =\int_{D-\{0\}}|K| \mathrm{e}^{2 \psi} \mathrm{~d} x \\ & \leqslant\left(\int_{D-\{0\}} K^{2} \mathrm{e}^{2 \psi} \mathrm{~d} x\right)^{1 / 2}\left(\int_{D-\{0\}} \mathrm{e}^{2 \psi} \mathrm{~d} x\right)^{1 / 2}, \end{aligned} $

i.e. $\Delta \psi \in L^{1}(D)$. Then we define the Newtonian potential of $\Delta \psi$ on $D$

$ \begin{aligned} v(x)=\int_{D} \mathit{\Gamma}(x-y) \Delta \psi(y) \mathrm{d} y. \end{aligned} $

From Theorem B, we have

$|x|^{2} \Delta \psi(x)=-|x|^{2} K(x) \mathrm{e}^{2 \psi(x)} \rightarrow 0 \quad \text { as } |x| \rightarrow 0. $

Applying Lemma 2.3, we obtain

$ \begin{aligned} \lim _{|x| \rightarrow 0} \frac{v(x)}{\ln |x|}=0 \quad \text { and } \quad \lim _{r \rightarrow 0} \operatorname{Osc}_{\theta}\{v(r, \theta)\}=0 . \end{aligned} $

Let $u=\psi-v$. By Corollary 1.1 with $f=\Delta \psi, v$ is smooth on $D-\{0\}$ and $\Delta v=\Delta \psi$. Then it follows that

$ \Delta u=0 \quad \text { on } D-\{0\}. $

Then from Proposition 1.1 in Ref.[3], we have

$ \begin{aligned} \lim _{|x| \rightarrow 0} \frac{u(x)}{\ln |x|}=\lim _{|x| \rightarrow 0} \frac{\psi(x)}{\ln |x|}=\alpha-1 . \end{aligned} $

Take $r <1$ small enough in $B_{r}$. Let $\bar{u}(x)=u(x)- (\alpha-1) \ln |x|$. By the Poisson integral, one can find $\hat{u}$ a smooth harmonic function on the $B_{r}$ that agrees with $\bar{u}$ on $\partial B_{r}$. We want to show that $\bar{u}=\hat{u}$ on $B_{r}-\{0\}$.

For $\varepsilon>0$, consider

$ w(x)=\bar{u}(x)-\hat{u}(x)+\varepsilon \ln |x| $

which is harmonic on $B_{r}-\{0\}$ and satisfies

$ \begin{aligned} \lim _{x \rightarrow 0} w(x)=-\infty. \end{aligned} $

By the maximum principle: for arbitrary small δ>0,

$ \begin{aligned} \sup _{B_{r}-\bar{B}_{\delta}} w=\sup _{\partial B_{r}} w=\varepsilon \sup _{\partial B_{r}} \ln |x| . \end{aligned} $

Taking $\delta \rightarrow 0$, we have: for any $\begin{aligned}x \in B_{r}-\{0\}\end{aligned}$, $\bar{u}(x)-\hat{u}(x)+\varepsilon \ln |x| \leqslant \sup\limits_{x \in B_r - \{0\}} w(x)=\varepsilon \sup\limits_{\partial B_{r}} \ln |x|$. Then taking $\varepsilon \rightarrow 0$, we obtain that $\bar{u}(x)-\hat{u}(x) \leqslant$ 0. The similar argument applies to $w(x)=\bar{u}(x)- \hat{u}(x)-\varepsilon \ln |x|$ and shows that $\bar{u}(x)-\hat{u}(x) \geqslant 0$ for every $x \in B_{r}-\{0\}$. Thus, 0 is a removable singularity for $\bar{u}$, i.e.for any $x \in D-\{0\}$,

$ u(x)=(\alpha-1) \ln |x|+\hat{u}(x) . $

In particular, we have

$ \begin{aligned} \lim _{r \rightarrow 0} \operatorname{Osc}_{\theta}\{u(r, \theta)\}=\lim _{r \rightarrow 0} \operatorname{Osc}_{\theta}\{\hat{u}(r, \theta)\}=0 . \end{aligned} $

Then we get as r→0,

$ \operatorname{Osc}_{\theta}\{\psi(r, \theta)\} \leqslant \operatorname{Osc}_{\theta}\{u(r, \theta)\}+\operatorname{Osc}_{\theta}\{v(r, \theta)\} \rightarrow 0 . $

(12)

Define a function $g(t)$ on $\mathbb{R}$ given by

$ g(t)=\left\{\begin{array}{l} 0, t \leqslant 0, \\ \operatorname{Osc}_{\theta}\{\psi(t, \theta)\}, t>0 . \end{array}\right. $

By (12), it is clear that g is continuous at 0. Then by Proposition 1.1 with $\mathit{Ω}=\left[\mathrm{e}^{-1}, \mathrm{e}^{1}\right]$, one can draw the conclusion.

It was showed that x=0 is either a weak cusp or a weak conical singular point with angle 2πα>0 in Ref. [2], which is expressed as the following.

Lemma 2.5 If $g=e^{2 \psi}|\mathrm{~d} z|^{2}$ is a metric in a punctured disk D-{0} with finite energy and area, then letting $r=|z|$ the following limit exists

$ \begin{aligned} \lim _{r \rightarrow 0} \frac{1}{2 \pi} \int_{0}^{2 \pi} \frac{\partial \psi}{\partial r}(r, \theta) \cdot r \mathrm{~d} \theta=\alpha-1 . \end{aligned} $

In particular, the singular point is called a cusp if $\alpha=0$.

With plenty of preparation, we now give the proof of Theorem A.

Proof of Theorem A As in Theorem B, it suffices to prove that for any sequence $\left\{x_{i}\right\} \rightarrow 0$,

$ \begin{aligned} \lim _{\left|x_{i}\right| \rightarrow 0}\left|x_{i}\right|\left|K\left(x_{i}\right)\right| \mathrm{e}^{\psi\left(x_{i}\right)}=0. \end{aligned} $

From the proof of Theorem B, we have

$ \begin{aligned} \lim _{i \rightarrow \infty}\left|K\left(x_{i}\right)\right| \mathrm{e}^{a_{i}}=0, \end{aligned} $

(13)

where $a_{i}=\inf _{y \in T_{1}} \psi_{i}(y)=\inf _{y \in T_{1}}\left(\psi\left(\left|x_{i}\right| y\right)+\right. \left.\ln \left|x_{i}\right|\right)$. Let

$ \widetilde{\psi}_{i}(y)=\psi\left(\left|x_{i}\right| y\right)+(1-\alpha) \ln \left(\left|x_{i}\right||y|\right) \quad \text { on } T_{2}. $

By Lemma 2.5, the function g is continuous at 0, which is given by

$ g(t)=\left\{\begin{array}{l} 0, t \leqslant 0, \\ \frac{1}{2 \pi} \int_{0}^{2 \pi}\left(t \frac{\partial \psi(t, \theta)}{\partial t}+1-\alpha\right) \mathrm{d} \theta, t>0. \end{array}\right. $

Let $r^{\prime}=\left|x_{i}\right| r, \mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$(in the sequel, the range of $r$ will always be $\left[\mathrm{e}^{-1}, \mathrm{e}^{1}\right]$ if it is omitted). Then by Proposition 1.1 with $\mathit{Ω}=\left[\mathrm{e}^{-1}, \mathrm{e}^{1}\right]$, as $i \rightarrow \infty$,

$ \frac{1}{2 \pi} \int_{0}^{2 \pi}\left(r^{\prime} \frac{\partial \psi\left(r^{\prime}, \theta\right)}{\partial r^{\prime}}+1-\alpha\right) \mathrm{d} \theta \rightarrow 0, $

uniformly for $\mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$. Since

$ \int_{0}^{2 \pi}\left(r^{\prime} \frac{\partial \psi\left(r^{\prime}, \theta\right)}{\partial r^{\prime}}+1-\alpha\right) \mathrm{d} \theta=\int_{0}^{2 \pi} r \frac{\partial \widetilde{\psi}_{i}(r, \theta)}{\partial r} \mathrm{~d} \theta, $

it follows that

$ \begin{aligned} \lim _{i \rightarrow \infty} \frac{1}{2 \pi} \int_{0}^{2 \pi} r \frac{\partial \widetilde{\psi}_{i}(r, \theta)}{\partial r} \mathrm{~d} \theta=0, \end{aligned} $

uniformly for $\mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$. It implies that

$ \begin{aligned} \lim _{i \rightarrow \infty} \frac{\partial}{\partial r} \int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta=0, \end{aligned} $

(14)

uniformly for $\mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$. In particular, we have

$ \begin{aligned} & \operatorname{Osc}_r\left\{\int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta\right\} \\ = & \max _r\left\{\int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta\right\}-\min _r\left\{\int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta\right\} \\ \leqslant & \sup _r\left\{\frac{\partial}{\partial r} \int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta\right\} \cdot 2 \mathrm{e} . \end{aligned} $

By (14), as $i \rightarrow \infty$,

$ \operatorname{Osc}_{r}\left\{\int_{0}^{2 \pi} \widetilde{\psi}_{i}(r, \theta) \mathrm{d} \theta\right\} \rightarrow 0 . $

(15)

For any $\theta_{0} \in[0, 2 \pi)$ and each i, denote

$ \begin{gathered} \widetilde{\psi}_i\left(r, \theta_r^i\right)=\frac{1}{2 \pi} \int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta, \forall \mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^1, \\ \widetilde{\psi}_i\left(\hat{r}_i, \theta_0\right)=\sup _r \widetilde{\psi}_i\left(r, \theta_0\right) \quad \text { and } \\ \widetilde{\psi}_i\left(\check{r}_i, \theta_0\right)=\underset{r}{\inf } \widetilde{\psi}_i\left(r, \theta_0\right) . \end{gathered} $

From Lemma 2.4, we know that

$ \begin{aligned} \lim _{i \rightarrow \infty} \operatorname{Osc}_{\theta}\left\{\widetilde{\psi}_{i}(r, \theta)\right\}=0 . \end{aligned} $

(16)

uniformly for $\mathrm{e}^{-1} \leqslant r \leqslant \mathrm{e}^{1}$. Since

$ \begin{aligned} & \operatorname{Osc}_r\left\{\widetilde{\psi}_i\left(r, \theta_0\right)\right\} \\ &=\left(\widetilde{\psi}_i\left(\hat{r}, \theta_0\right)-\widetilde{\psi}_i\left(\hat{r}, \theta_{\hat{r}}^i\right)\right)+\left(\widetilde{\psi}_i\left(\check{r}, \theta_r^i\right)-\widetilde{\psi}_i\left(\check{r}, \theta_0\right)\right)+ \\ &\left(\widetilde{\psi}_i\left(\hat{r}, \theta_{\hat{r}}^i\right)-\widetilde{\psi}_i\left(\check{r}, \theta_r^i\right)\right) \\ & \leqslant \operatorname{Osc}_\theta\left\{\widetilde{\psi}_i(\hat{r}, \theta)\right\}+\operatorname{Osc}_\theta\left\{\widetilde{\psi}_i\left(\check{r}, \theta_0\right)\right\}+ \\ & \frac{1}{2 \pi} \operatorname{Osc}_r\left\{\int_0^{2 \pi} \widetilde{\psi}_i(r, \theta) \mathrm{d} \theta\right\}, \end{aligned} $

Combined (15), we have: as $i \rightarrow \infty$,

$ \operatorname{Osc}_{r}\left\{\widetilde{\psi}_{i}\left(r, \theta_{0}\right)\right\} \rightarrow 0 . $

(17)

Furthermore, for each i, denote

$ \widetilde{\psi}_{i}\left(r_{1}, \theta_{1}\right)=\max _{(r, \theta) \in \bar{T}_{1}}\left\{\widetilde{\psi}_{i}(r, \theta)\right\} $

and

$ \widetilde{\psi}_{i}\left(r_{2}, \theta_{2}\right)=\min _{(r, \theta) \in \bar{T}_{1}}\left\{\widetilde{\psi}_{i}(r, \theta)\right\}, $

where $r_{1}, \theta_{1}, r_{2}, \theta_{2}$ vary with different $i$. Since for each $i$,

$ \begin{aligned} & \operatorname{Osc}_{r, \theta \in \bar{T}_1}\left\{\widetilde{\psi}_i(r, \theta)\right\} \\ = & \widetilde{\psi}_i\left(r_1, \theta_1\right)-\widetilde{\psi}_i\left(r_2, \theta_2\right) \\ = & \widetilde{\psi}_i\left(r_1, \theta_1\right)-\widetilde{\psi}_i\left(r_1, \theta_0\right)+\widetilde{\psi}_i\left(r_2, \theta_0\right)-\widetilde{\psi}_i\left(r_2, \theta_2\right)+ \\ & \widetilde{\psi}_i\left(r_1, \theta_0\right)-\widetilde{\psi}_i\left(r_2, \theta_0\right) \\ \leqslant & \operatorname{Osc}_\theta\left\{\widetilde{\psi}_i\left(r_1, \theta\right)\right\}+\operatorname{Osc}_\theta\left\{\widetilde{\psi}_i\left(r_2, \theta\right)\right\}+\operatorname{Osc}_r\left\{\widetilde{\psi}_i\left(r, \theta_0\right)\right\}, \end{aligned} $

by (16) and (17), it follows that

$ \begin{aligned} \lim _{i \rightarrow \infty} \operatorname{Osc}_{(r, \theta) \in \bar{T}_{1}}\left\{\widetilde{\psi}_{i}(r, \theta)\right\}=0 . \end{aligned} $

Consequently,

$ \begin{aligned} \operatorname{Osc}_{y \in T_{1}} \psi_{i}(y) & =\operatorname{Osc}_{y \in T_{1}}\left\{\widetilde{\psi}_{i}(y)+\alpha \ln \left(\left|x_{i}\right||y|\right)-\ln |y|\right\} \\ & \leqslant 3(\alpha+1), \end{aligned} $

when i is sufficiently large. In view of (5) and the above inequalities, we get

$ \begin{aligned} \lim _{\left|x_{i}\right| \rightarrow 0}\left|x_{i}\right|\left|K\left(x_{i}\right)\right| \mathrm{e}^{\psi\left(x_{i}\right)} & \leqslant \lim _{\left|x_{i}\right| 0}\left|K\left(x_{i}\right)\right|\left|\sup _{y \in T_{1}} \mathrm{e}^{\psi_{i}(y)}\right| \\ & \leqslant \lim _{\mid x_{i} \nmid 0} \mathrm{e}^{3(\alpha+1)}\left|K\left(x_{i}\right)\right| \mathrm{e}^{\alpha_{i}}=0 . \end{aligned} $

3 Computation of an example

In this section, we will give an example to illustrate Theorem A. The HCMU metric is a special case of the extremal Hermitian metric. Chen et al.^[10-11] have studied the HCMU metrics. They showed one can use the following method to construct HCMU metrics on a compact Riemann surface M. First one can consider the following equation on M

$ \left\{\begin{array}{l} \frac{\mathrm{d} K}{-\frac{1}{3}\left(K-K_{1}\right)\left(K-K_{2}\right)\left(K+K_{1}+K_{2}\right)}=\omega+\bar{\omega}, \\ K\left(p_{0}\right)=K_{0}, \end{array}\right. $

(18)

where ω is a meromorphic 1-form on M satisfying:

1) ω only has simple poles,

2) the residues of ω at poles are all real numbers,

3) $\omega+\bar{\omega}$ is exact on $M-\{$ poles and zeros of $\omega\}$; $K_{1}$ and $K_{2}$ are real numbers which satisfy $K_{1}>0$ and $-\frac{K_{1}}{2} \leqslant K_{2} <K_{1} ; p_{0}$ is a point in $M$－$\{$ poles and zeros of $\omega\}$ and $K_{2} <K_{0} <K_{1}$. Then one can show(18) has a unique smooth solution $K$ on $M$－$\{$ poles and zeros of $\omega\}$ and $K$ can be continuously extended to $M$. Furthermore $K_{2} <K <K_{1}$ on $M$－$\{$ poles of $\omega\}, K$ takes the maximum $K_{1}$ at the poles at which the residue of $\omega$ is negative and $K$ takes the minimum $K_{2}$ at the poles at which the residue of $\omega$ is positive. Then we can define a metric $g$ on $M$－{ poles and zeros of ω } by

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

(19)

One can prove g is an HCMU metric with finite area and finite energy and K is the Gauss curvature of g. Furthermore, the set of the singularities of g is a subset of the set of zeros and poles of ω. If $K_{2}> -\frac{K_{1}}{2}$, there are three cases about the singularities of g:

1) if p is a zero of ω, then p is a conical singularity with the conical angle $2 \pi\left(\operatorname{ord}_{p} \omega+1\right)$;

2) denote $-\frac{3}{\left(K_{1}-K_{2}\right)\left(K_{2}+2 K_{1}\right)}$ by $\sigma$. If $p$ is a pole of $\omega$ with the negative residue of $\omega$ and $\operatorname{Res}_{p}(\omega) \neq \sigma$, then $p$ is a conical singularity with the conical angle $2 \pi \frac{\operatorname{Res}_{p}(\omega)}{\sigma}$;

3) denote $-\frac{2 K_{1}+K_{2}}{K_{1}+2 K_{2}}$ by $\tau$. If $p$ is a pole of $\omega$ with the positive residue of $\omega$ and $\operatorname{Res}_{p}(\omega) \neq \sigma \tau$, then $p$ is a conical singularity with the conical angle $2 \pi \frac{\operatorname{Res}_{p}(\omega)}{\sigma \tau}$.

If $K_{2}=-\frac{K_{1}}{2}$, there are also three cases about the singularities of $g$ :

1) if $p$ is a zero of $\omega$, then $p$ is a conical singularity with the conical angle $2 \pi\left(\operatorname{ord}_{p} \omega+1\right)$;

2) denote $-\frac{3}{\left(K_{1}-K_{2}\right)\left(K_{2}+2 K_{1}\right)}$ by $\sigma$. If $p$ is a pole of $\omega$ with the negative residue of $\omega$ and $\operatorname{Res}_{p}(\omega) \neq \sigma$, then $p$ is a conical singularity with the conical angle $2 \pi \frac{\operatorname{Res}_{p}(\omega)}{\sigma}$;

3) if p is a pole of ω with the positive residue of ω, p is a cusp.

Next, we calculate the following limit

$ \begin{aligned} \lim _{z \rightarrow 0}|z| K e^{\psi}, \end{aligned} $

at any singularity of g.

If p is a conical singularity of g with the conical angle 2πα, then in a neighborhood of p

$ g=h|z|^{2 \alpha-2}|\mathrm{~d} z|^{2}=\mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2}, $

where z is the local coordinate in a neighborhood of p with $z(p)=0$ and $h$ is a continuous function in the neighborhood with $h(0) \neq 0$. Thus $\mathrm{e}^{2 \psi}=h|z|^{2 \alpha-2}$. Then

$ |z| K \mathrm{e}^{\psi}=|z| K \sqrt{h}|z|^{\alpha-1}=K \sqrt{h}|z|^{\alpha} . $

Since both K and $\sqrt{h}$ are continuous at p and $\alpha>0$,

$ \begin{aligned} \lim _{z \rightarrow 0}|z| K \mathrm{e}^{\psi}=\lim _{z \rightarrow 0} K \sqrt{h}|z|^{\alpha}=0 . \end{aligned} $

Therefore Theorem A holds in conical singularity case of HCMU metric.

If p is a cusp, then $K_{2}=-\frac{K_{1}}{2}$ and

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

Suppose $\omega=\frac{\hat{f}(z)}{z} \mathrm{~d} z$ in a neighborhood of p, where z is the local coordinate in the neighborhood of p with $z(p)=0$ and $\hat{f}(z)$ is a holomorphic function in the neighborhood with $\hat{f}(0) \neq 0$. Then in the neighborhood of $p$

$ \begin{gathered} g=-\frac{4}{3}\left(K-K_{1}\right)\left(K-K_{2}\right)^{2} \frac{|\hat{f}(z)|^{2}}{|z|^{2}}|\mathrm{~d} z|^{2}= \\ \mathrm{e}^{2 \psi}|\mathrm{~d} z|^{2} . \end{gathered} $

Therefore

$ \mathrm{e}^{\psi}=\sqrt{-\frac{4}{3}\left(K-K_{1}\right)}\left(K-K_{2}\right) \frac{|\hat{f}(z)|}{|z|} . $

Then

$ |z| K \mathrm{e}^{\psi}=K \sqrt{-\frac{4}{3}\left(K-K_{1}\right)}\left(K-K_{2}\right)|\hat{f}(z)| . $

Since K is continuous at p with $K(p)=K_{2}$, $\lim _{z \rightarrow 0}=|z| K \mathrm{e}^{\psi}$

$ \begin{aligned} =\lim _{z \rightarrow 0} K \sqrt{-\frac{4}{3}\left(K-K_{1}\right)}\left(K-K_{2}\right)|\hat{f}(z)|=0 . \end{aligned} $

Therefore Theorem A holds in cusp case of HCMU metric.