text:
\section{Introduction}
Let $(\mathbb{S}^2,g_0)$ be the 2-dimensional sphere $x_1^2+x_2^2+x_3^2=1$ with the metric $g_0=dx_1^2+dx_2^2+dx_3^2$
and the corresponding volume element $dv_{g_0}$. It was proved by Moser \cite{Moser}, via a symmetrization argument, that
there exists some constant $C$ such that if
$u$ satisfies $\int_{\mathbb{S}^2}|\nabla_{g_0} u|^2dv_{g_0}\leq 1$ and $\int_{\mathbb{S}^2}udv_{g_0}=0$, then
\begin{equation}\label{Mosr-2}\int_{\mathbb{S}^2} e^{\alpha u^2}dv_{g_0}\leq C,\quad\forall\alpha\leq 4\pi;\end{equation}
moreover, $4\pi$ is the best constant in the sense that if $\alpha>4\pi$, then the integerals in (\ref{Mosr-2}) are still finite,
but there is no uniform constant $C$ such that (\ref{Mosr-2}) holds.
Also, it was pointed out by Moser \cite{Moser-73} that the same argument in \cite{Moser} indicates a similar inequality for even functions.
 Namely, there exists a constant $C$ such that if $u$ satisfies $u(-x)=u(x)$ for all $x\in \mathbb{S}^2$, $\int_{\mathbb{S}^2}|\nabla_{g_0}u|^2dv_{g_0}\leq 1$,
and $\int_{\mathbb{S}^2}udv_{g_0}=0$, then
\begin{equation}\label{Mosr-even}
\int_{\mathbb{S}^2} e^{\alpha u^2}dv_{g_0}\leq C,\quad\forall\alpha\leq 8\pi.\end{equation}
Observing that the inequality (\ref{Mosr-even}) can be applied to the prescribing Gaussian curvature equation, Moser \cite{Moser-73}
obtained the following: If $K:\mathbb{S}^2\rightarrow \mathbb{R}$ is a smooth function satisfying $K(-x)=K(x)$ for all $x\in\mathbb{S}^2$, then there is a
smooth solution to
\begin{equation}\label{pres-S2}\Delta_{g_0}u+1-Ke^{2u}=0,\end{equation}
where $\Delta_{g_0}$ is the Laplace-Beltrami operator. The geometric meaning of the above equation is as follows: If $u$ is a smooth solution to
(\ref{pres-S2}), then the metric $g=e^{2u}g_0$ has the Gaussian curvature $K$.

Moser's inequality (\ref{Mosr-2}) was generalized by Fontana \cite{Fontana} to Riemannian manifolds case. In particular, let $(\Sigma,g)$ be a
closed Riemannian surface. Then for all smooth functions $u$ with $\int_\Sigma|\nabla_gu|^2dv_g\leq 1$ and $\int_\Sigma udv_g=0$, there exists
a constant depending only on $(\Sigma,g)$ such that
$$\int_\Sigma e^{4\pi u^2}dv_g\leq C;$$
moreover, $4\pi$ is the best constant for the above inequality.
Also Moser's inequality (\ref{Mosr-even}) was generalized to various versions. Let $\textbf{G}=
\{\sigma_1,\cdots,\sigma_N\}$ be an isometric group acting on a closed Riemannian surface $(\Sigma,g)$, $I(x)$ be the number of all distinct elements in $\textbf{G}(x)=\{\sigma_1(x),\cdots,\sigma_N(x)\}$, and $\ell=\inf_{x\in\Sigma}I(x)$. Using isoperimetric inequalities, among other results Chen \cite{Chen-90} proved that there exists a constant $C$ such that for all smooth functions $u$ with
$u(\sigma(x))=u(x)$ for all $\sigma\in\textbf{G}$ and all $x\in\Sigma$, $\int_\Sigma|\nabla_gu|^2dv_g\leq 1$ and $\int_\Sigma udv_g=0$, there holds
\begin{equation}\label{Chen}\int_\Sigma e^{4\pi\ell u^2}dv_g\leq C.\end{equation}
Recently, motivated by \cite{Yang-Tran, Lu-Yang-1, Yang-JDE-15}, Fang-Yang \cite{Fang-Yang} employed blow-up analysis to improve (\ref{Chen}) to
analogous inequalities involving eigenvalues of the Laplace-Beltrami operator $\Delta_g$.

Another famous equation similar to the prescribing curvature equation (\ref{pres-S2}) is the  mean field equation, namely
\begin{equation}\label{meanfield}\Delta_g u=\rho\le(\frac{he^u}{\int_\Sigma he^udv_g}-\frac{1}{{\rm Vol}_g(\Sigma)}\right),\end{equation}
where $\rho$ is some real number. When $h$ is a smooth positive function, it was proved by Ding-Jost-Li-Wang \cite{DJLW} that for $\rho<8\pi$, (\ref{meanfield}) has a solution;
under certain geometric condition on $(\Sigma,g)$, (\ref{meanfield}) has a solution for $\rho=8\pi$ (when $\Sigma$ is a flat torus, this result was independently proved by
Nolasco-Tarantello \cite{NT}). The authors \cite{Yang-Zhu-Proc} derived the same conclusion when $h\geq 0$ and $h\not\equiv 0$.
Struwe-Tarantello \cite{Struwe} obtained a non-constant solution of (\ref{meanfield}), when $\Sigma$ is a
flat torus, $h\equiv1$ and $\rho\in(8\pi,4\pi^2)$.
When $\Sigma$ is a compact Riemannian surface of positive genus and $h$ is a smooth positive function,
Ding-Jost-Li-Wang \cite{DJLW2} proved that (\ref{meanfield}) admits a non-minimal solution for $\rho\in(8\pi,16\pi)$.
This result was generalized first by Chen-Lin \cite{ChenLin1, ChenLin2} to $\rho\in(8m\pi,16m\pi)$ ($m\in\mathbb{Z}^+$), then
by Malchiodi \cite{Malchiodi} to $\rho\in(8m\pi,16m\pi)$ ($m\in\mathbb{Z}^+$) and $\Sigma$ is a general Riemannian surface. Let $K$ be the Gaussian
curvature, Chen-Lin \cite{ChenLin1} also proved that, if $\Delta_g\log h(x)+\frac{8m\pi}{{\rm Vol}_g(\Sigma)}-2K(x)>0$ for any $x\in\Sigma$,
then (\ref{meanfield}) has a solution. This improved the result in \cite{DJLW}.
By assuming $\Sigma$ to be a
flat torus and $h$ a
positive smooth function with certain symmetrization, Wang \cite{Wang} obtained an analog of \cite{DJLW}
for $\rho=16\pi$.

Our aim is to extend Ding-Jost-Li-Wang's result \cite{DJLW} to closed Riemannian surface with an isometric group action.
More precisely, let $(\Sigma,g)$, ${\textbf G}$ and $\ell$ be defined as in (\ref{Chen}). If $h$ is a smooth positive function
and $h(\sigma(x))=h(x)$ for all $\sigma\in
\textbf{G}$ and all $x\in\Sigma$, then for $\rho<8\pi\ell$, (\ref{meanfield}) has a solution; for $\rho=8\pi\ell$, under certain geometric hypothesis,
(\ref{meanfield}) has a solution.

\section{Notations and main results}

Let $(\Sigma,g)$ be a closed Riemannian surface and
$\textbf{G}=\{\sigma_1,\cdots,\sigma_N\}$ be
a finite isometric group acting on it. By definition, $\textbf{G}$ is a group and each $\sigma_i:\Sigma\rightarrow\Sigma$
is an isometric map, particularly $\sigma_i^\ast g_x=g_{\sigma_i(x)}$ for all $x\in \Sigma$.
Let  $u:\Sigma\rightarrow\mathbb{R}$ be a measurable function, we say that $u\in\mathscr{I}_{\textbf{G}}$ if $u$ is $\textbf{G}$-invariant,
namely $u(\sigma_i(x))=u(x)$ for any $1\leq i\leq N$ and almost every $x\in\Sigma$.
We denote $W^{1,2}(\Sigma,g)$
the closure of $C^\infty(\Sigma)$ under the norm
$$\|u\|_{W^{1,2}(\Sigma,g)}=\le(\int_\Sigma\le(|\nabla_gu|^2+u^2\right)dv_g\right)^{1/2},$$
where $\nabla_g$ and $dv_g$ stand for the gradient operator and the Riemannian volume element respectively.
Define a Hilbert space
\begin{equation}\label{HG}\mathscr{H}_\textbf{G}= \le\{u\in W^{1,2}(\Sigma,g)\cap \mathscr{I}_\textbf{G}: \int_\Sigma udv_g=0\right\}\end{equation}
with an inner product
$$\langle u,v\rangle_{\mathscr{H}_{\textbf{G}}}=\int_\Sigma \langle\nabla_gu,\nabla_gv\rangle dv_g,$$
where $\langle\nabla_gu,\nabla_gv\rangle$ stands for the Riemannian inner product of $\nabla_gu$ and $\nabla_gv$.
Let $\Delta_g=-{\rm div}_g\nabla_g$ be the Laplace-Beltrami operator. For any $x\in\Sigma$, we set $I(x)=\sharp \textbf{G}(x)$, where $\sharp A$
stands for the number of all distinct points in the set $A$, and $\textbf{G}(x)=\{\sigma_1(x),\cdots,\sigma_N(x)\}$.
 Let
\begin{equation}\label{ell}\ell=\inf_{x\in\Sigma}I(x).\end{equation}
Clearly we have $1\leq\ell\leq N$ since $1\leq I(x)\leq N$ for all $x\in\Sigma$.
Chen's inequality (\ref{Chen}) is equivalent to the following: $\forall \beta\leq 4\pi\ell$, there holds
\begin{equation}\label{T-M-ineq}\sup_{u\in\mathscr{H}_\textbf{G},\,\int_\Sigma|\nabla_gu|^2dv_g\leq 1}\int_\Sigma e^{\beta u^2}dv_g<\infty.\end{equation}
Define a functional on $\mathscr{H}_\textbf{G}$ by
\begin{equation}\label{functional}J_\gamma(u)=\frac{1}{2}\int_\Sigma|\nabla_gu|^2dv_g-\gamma\log\int_\Sigma he^udv_g.\nonumber\end{equation}
For any $0<\epsilon<1$, it follows from (\ref{T-M-ineq}) and a direct method of  variation that the minimizer $u_\epsilon$ of
the subcritical functional $J_{8\pi\ell(1-\epsilon)}$ exists. Denote $c_\epsilon=u_\epsilon(x_\epsilon)=\max_{\Sigma}u_\epsilon$.
As we shall see, if $c_\epsilon$ is bounded, then up to
a subsequence,
$u_\epsilon$ would converge to some $u_0$ as $\epsilon$ tends to zero, and $u_0$ is a minimizer of the critical functional $J_{8\pi\ell}$;
if $u_\epsilon$ blows up, i.e. $c_\epsilon\rightarrow \infty$, then up to
a subsequence, $x_\epsilon$ converges to some $x_0$, $I(x_0)=\ell$, and  $u_\epsilon$ converges to certain Green function $\widetilde{G}_{x_0}$ satisfying
\begin{equation}\label{Greenfunct}\le\{\begin{array}{lll}\Delta_g\widetilde{G}_{x_0}=8\pi\sum_{i=1}^{\ell}\delta_{\sigma_i(x_0)}
-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\\[1.5ex] \int_\Sigma \widetilde{G}_{x_0}dv_g=0,\end{array}\right.\nonumber\end{equation}
where we assume without loss of generality $\textbf{G}(x_0)=\{\sigma_1(x_0),\cdots,\sigma_\ell(x_0)\}$.
In a normal coordinate system $\{x_1,x_2\}$ near $x_0$, the Green function $\widetilde{G}_{x_0}$ can be written as
\begin{equation}\label{G-repres}\widetilde{G}_{x_0}(x)=-4\log r+\widetilde{A}_{x_0}+b_1x_1+b_2x_2+c_1x_1^2+2c_2x_1x_2+c_3x_2^2+O(r^3),\end{equation}
where $\widetilde{A}_{x_0}$, $b_1$, $b_2$, $c_1$, $c_2$, $c_3$ are constants, $r(x)$ denotes the geodesic distance between $x$ and $x_0$.

Our first result reads

\begin{theorem}\label{Thm1} Let $(\Sigma,g)$ be a closed Riemannian surface, $\textbf{G}=\{\sigma_1,\cdots,\sigma_N\}$ be an isometric group acting on it,
and $\mathscr{H}_{\textbf{G}}$ be a function space defined as in (\ref{HG}). Let $\ell$ be defined as in (\ref{ell}).
Suppose $h$ is a smooth positive function on $\Sigma$
and $h(\sigma(x))=h(x)$ for all $\sigma\in\textbf{G}$ and all $x\in\Sigma$. Then
$(i)$ for any $\rho<8\pi\ell$, the equation (\ref{meanfield}) has a solution;
$(ii)$ if (\ref{meanfield}) has no solution for $\rho=8\pi\ell$, there holds
\begin{equation}\label{lower}\inf_{u\in\mathscr{H}_{\textbf{G}}}J_{8\pi\ell}(u)\geq -4\pi\ell\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)-8\pi\ell,\end{equation}
where $\widetilde{A}_x$ is defined as in  (\ref{G-repres}).
\end{theorem}

Noting that $J_{8\pi\ell}(0)\geq \inf_{u\in\mathscr{H}_{\textbf{G}}}J_{8\pi\ell}(u)$, we have the following:

\begin{theorem}\label{Thm2} Under the assumptions of Theorem \ref{Thm1}, if in addition
\begin{equation}\label{cond}\log\int_\Sigma hdv_g>1+\frac{1}{2}\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x),\end{equation}
 then (\ref{meanfield}) has a solution for $\rho=8\pi\ell$.
\end{theorem}

Later we shall give an example of $(\Sigma,g)$, $\textbf{G}$ and $h$ satisfying the hypothesis (\ref{cond}).
Assuming further $I(x)$ is a constant, we shall prove another existence result for the mean field equation (\ref{meanfield}),
namely

\begin{theorem}\label{Thm3}
Let $(\Sigma,g)$ be a closed Riemannian surface, $\textbf{G}=\{\sigma_1,\cdots,\sigma_\ell\}$ be an isometric group acting on it,
and $\mathscr{H}_{\textbf{G}}$ be a function space defined as in (\ref{HG}). Let $I(x)$ be defined as in (\ref{ell}).
Suppose that $I(x)\equiv \ell$ for all $x\in\Sigma$, that
$h$ is a smooth positive function satisfying $h(\sigma(x))=h(x)$ for all $\sigma\in \textbf{G}$ and all $x\in\Sigma$, that
$2\log(\pi\ell h(p))+\widetilde{A}_p=\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)$, and that in a normal coordinate system near $p$,
$$h(x)-h(p)=k_1x_1+k_2x_2+k_3x_1^2+2k_4x_1x_2+k_5x_2^2+O(r^3).$$
If in addition
\begin{equation}\label{hy-2}\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}-2K(p)+b_1^2+b_2^2-\frac{\Delta_g h(p)}{h(p)}+\frac{2(k_1b_1+k_2b_2)}{h(p)}>0,\end{equation}
then (\ref{meanfield}) has a solution for $\rho=8\pi\ell$, where $K(p)$ denotes the Gaussian curvature of $(\Sigma,g)$ at $p$.
\end{theorem}

In the case $\textbf{G}=\{Id\}$, where $Id:\Sigma\rightarrow \Sigma$ is the identity map, Theorems \ref{Thm1}-\ref{Thm3} are reduced to that of
Ding-Jost-Li-Wang \cite{DJLW}. Though we are in the spirit of Ding-Jost-Li-Wang \cite{DJLW} for the proof
of the general case of Theorems \ref{Thm1}-\ref{Thm3}, many technical difficulties need to be smoothed. The first issue is to construct
Green functions having many singular points by using elliptic estimates and the symmetric properties of $(\Sigma,g)$. The second one is to
derive a lower bound of $J_{8\pi\ell}$ on $\mathscr{H}_{\textbf{G}}$ by using the maximum principle. Here we use an argument of
our previous work \cite{Yang-Zhu-SCM}
which quite simplified that of \cite{DJLW}. The third issue is to construct test functions showing that (\ref{lower}) does not hold under
the hypothesis (\ref{hy-2}), which implies that (\ref{meanfield}) has a solution. For some technical reason we assume $I(x)$ is a constant
for all $x\in \Sigma$. Even in the case $\textbf{G}=\{Id\}$, our test function is different from that of Ding-Jost-Li-Wang \cite{DJLW}.\\

The remaining part of this paper is organized as follows: In Section \ref{Sec3}, we discuss existence and properties of
certain Green function in our setting; In Section \ref{Sec4}, we prove Theorems \ref{Thm1} and \ref{Thm2}; In Section \ref{Sec5},
we prove Theorem \ref{Thm3}; In Section \ref{Sec6}, we give examples of surfaces and isometric groups satisfying the hypotheses of
Theorems \ref{Thm2} and \ref{Thm3}. Throughout this paper, we do not distinguish sequence and subsequence.

\section{Green functions}\label{Sec3}
In this section, we discuss Green functions defined on a closed Riemannian surface $(\Sigma,g)$ with the action of an isometric group
$\textbf{G}=\{\sigma_1,\cdots,\sigma_N\}$.
Let $G(x,y)$ be the  usual Green function on $(\Sigma,g)$, which is a distributional solution to
\begin{equation}\label{Gree}\Delta_{g,y}G(x,y)=8\pi\delta_x(y)-\frac{8\pi}{{\rm Vol}_g(\Sigma)},\end{equation}
where $\Delta_{g,y}$ is the Laplace-Beltrami operator with respect to the variable $y$, $\delta_x$ is the standard Dirac measure centered
at $x$.
It follows from (\cite{Aubin}, Theorem 4.13) that such a $G$ does exist and
can be normalized so that $\int_\Sigma G(x,y)dv_{g,y}=0$ for all $x\in \Sigma$. Moreover, $G(x,y)=G(y,x)$ and
$G\in C^\infty(\Sigma\times\Sigma\setminus\{(x,x): x\in\Sigma\})$. A key observation is the following:

\begin{proposition}\label{prop-1}
For all $\sigma\in\textbf{G}$ and all $x,y\in\Sigma$, $x\not=y$, there holds $G(\sigma(x),\sigma(y))=G(x,y)$.
\end{proposition}

\proof Take $\sigma\in\textbf{G}$. Replacing $x$ and $y$ by two points $\sigma(x)$ and $\sigma(y)$ in (\ref{Gree}), we have that
$$\Delta_{g,\,\sigma(y)}G(\sigma(x),\sigma(y))=8\pi\delta_{\sigma(x)}(\sigma(y))-\frac{8\pi}{{\rm Vol}_g(\Sigma)}.$$
Since $\Delta_{g,\sigma(y)}=\Delta_{g,y}$ and $\delta_{\sigma(x)}(\sigma(y))=\delta_{x}(y)$, there holds  in the distributional sense
$$\Delta_{g,y}\le(G(\sigma(x),\sigma(y))-G(x,y)\right)=0.$$
Hence $G(\sigma(x),\sigma(y))-G(x,y)\equiv C$ for some constant $C$. Since
$$\int_\Sigma G(\sigma(x),\sigma(y))dv_{g,\,\sigma(y)}=\int_\Sigma G(x,y)dv_{g,y}=0,$$
we conclude $C=0$ and the proposition. $\hfill\Box$\\

Given any $x\in\Sigma$. Assume that $I(x)=\sharp\textbf{G}(x)=
\sharp\{\sigma_1(x),\cdots,\sigma_N(x)\}=j$. Recall the definition (\ref{ell}),
$\ell=\inf_{x\in\Sigma}I(x)$. It follows that $\ell\leq j\leq N$. With no loss of generality, we can assume that
\begin{equation}\label{tilde-j}\textbf{G}(x)=\{\sigma_1(x),\cdots,\sigma_j(x)\}.\end{equation}
Then we have the following existence result:
\begin{proposition}\label{prop-2} Let $x\in \Sigma$ be fixed such that $I(x)=j$ and (\ref{tilde-j}) holds. Then
there exists a unique Green function $\widetilde{G}_x$ satisfying
$$\le\{\begin{array}{lll}\Delta_g\widetilde{G}_x=8\pi\sum_{i=1}^j\delta_{\sigma_i(x)}-\frac{8\pi j}{{\rm Vol}_g(\Sigma)}\\[1.5ex]
\int_\Sigma \widetilde{G}_xdv_g=0.\end{array}\right.$$
Moreover, $\widetilde{G}_x$ can be explicitly written as
\begin{equation}\label{Gtilde}\widetilde{G}_x(y)=G(\sigma_1(x),y)+\cdots+G(\sigma_j(x),y),\nonumber\end{equation}
where $G(x,y)$ be a solution of (\ref{Gree}) with $\int_\Sigma G(x,y)dv_{g,y}=0$.
\end{proposition}
\proof Note that in the distributional sense
$$\Delta_{g,y}\le(\widetilde{G}_x(y)-G(\sigma_1(x),y)-\cdots-G(\sigma_j(x),y)\right)=0.$$
It follows that $\widetilde{G}_x(y)-G(\sigma_1(x),y)-\cdots-G(\sigma_j(x),y)\equiv C$ for some constant $C$.
Since $$\int_\Sigma \widetilde{G}_x(y) dv_{g,y}=\int_\Sigma {G}(\sigma_i(x),y)dv_{g,y}=0,\quad i=1,\cdots,j,$$
we have $C=0$, which is the desired result. $\hfill\Box$ \\

By elliptic estimates, one writes near $x$
\begin{equation}\label{g-dec}G(x,y)=-4\log r+A_x+O(r),\end{equation}
where $r$ denotes the geodesic distance between $x$ and $y$, $A_x=\lim_{r\rightarrow 0}(G(x,y)+4\log r)$. In the same way, one can
write near $\sigma_i(x)$ \begin{equation}\label{tild-g}\widetilde{G}_x=-4\log r+\widetilde{A}_{\sigma_i(x)}+O(r),\end{equation}
where $r$ denotes the geodesic distance between $\sigma_i(x)$ and $y$, $\widetilde{A}_{\sigma_i(x)}=\lim_{r\rightarrow 0}(\widetilde{G}_x(y)+4\log r)$.
Another key observation is the following:

\begin{proposition}\label{prop-3}
Let $x\in\Sigma$ be such that $I(x)=j$ and (\ref{tilde-j}) be satisfied. Assuming (\ref{g-dec}) and (\ref{tild-g}), we have  $\widetilde{A}_{\sigma_i(x)}=\widetilde{A}_{\sigma_1(x)}$, $i=1,\cdots,j$.
\end{proposition}

\proof By Proposition \ref{prop-2},
$$\widetilde{G}_x(y)=G(\sigma_1(x),y)+\cdots+G(\sigma_j(x),y).$$
Clearly
\begin{equation}\label{s-1}
\widetilde{A}_{\sigma_i(x)}=A_{\sigma_i(x)}+\sum_{k\not=i,\,k=1}^jG(\sigma_i(x),\sigma_k(x)).
\end{equation}
By Proposition \ref{prop-1}, we can see that
\begin{eqnarray}\nonumber
A_{\sigma_i(x)}&=&\lim_{r\rightarrow 0}(G(\sigma_i(x),y)+4\log r)\\\nonumber
&=&\lim_{r\rightarrow 0}(G(\sigma_i(x),\sigma_i(z))+4\log r)\\\nonumber
&=&\lim_{r\rightarrow 0}(G(x,z)+4\log r)\\\label{s-2}
&=&A_{\sigma_1(x)}.
\end{eqnarray}
Here in the second equality, $r$ denotes the geodesic distance between $\sigma_i(x)$ and $\sigma_i(z)$.
While in the third equality, $r$  denotes the geodesic distance between $x$ and $z$.
Since $G(\sigma_i(x),\sigma_k(x))=G(\sigma_k(x),\sigma_i(x))$, we conclude
\begin{equation}\label{s-3}\sum_{k\not=i,\,k=1}^jG(\sigma_i(x),\sigma_k(x))=\sum_{k=2}^jG(\sigma_1(x),\sigma_k(x)),
\quad\forall i=1,\cdots,j.\end{equation}
Inserting (\ref{s-2}) and (\ref{s-3}) into (\ref{s-1}), we have $\widetilde{A}_{\sigma_i(x)}=\widetilde{A}_{\sigma_1(x)}$,
$i=1,\cdots,j$, as desired. $\hfill\Box$

\section{Proof of Theorems \ref{Thm1} and \ref{Thm2} }\label{Sec4}

\subsection{Proof of Theorem \ref{Thm1}}
Clearly $(i)$ of Theorem \ref{Thm1} is implied by the following lemma.
\begin{lemma}\label{sub-critical}
For any $0<\epsilon<1$, there exists some $u_\epsilon\in\mathscr{H}_\textbf{G}$ such that
\begin{equation}\label{sub}J_{8\pi\ell(1-\epsilon)}(u_\epsilon)=\inf_{u\in\mathscr{H}_\textbf{G}}J_{8\pi\ell(1-\epsilon)}(u).\end{equation}
Moreover, $u_\epsilon$ is a solution of the equation
\begin{equation}\label{E-L}\le\{\begin{array}{lll}
 \Delta_gu_\epsilon=8\pi\ell(1-\epsilon)\le(\lambda_\epsilon^{-1}he^{u_\epsilon}-\frac{1}{{\rm Vol}_g(\Sigma)}\right)\\[1.5ex]
 \lambda_\epsilon=\int_\Sigma he^{u_\epsilon}dv_g.
\end{array}\right.\end{equation}
\end{lemma}
\proof In view of (\ref{T-M-ineq}), a direct method of variation leads to the existence of $u_\epsilon\in \mathscr{H}_\textbf{G}$
satisfying (\ref{sub}). Then a straightforward calculation shows $u_\epsilon$ satisfies the Euler-Lagrange equation (\ref{E-L}). $\hfill\Box$\\

For the proof of $(ii)$ of Theorem \ref{Thm1}, we modify an argument of blow-up analysis performed by Ding-Jost-Li-Wang \cite{DJLW}.
Let $c_\epsilon=u_\epsilon(x_\epsilon)=\max_\Sigma u_\epsilon$. If $c_\epsilon$ is bounded, then applying elliptic estimates to
(\ref{E-L}), we conclude that up to a subsequence, $u_\epsilon\rightarrow u_0$ in $C^1(\Sigma,g)$. It is easy to see that
$u_0$ is a solution of (\ref{meanfield}) for $\rho=8\pi\ell$. Therefore, under the assumption of $(ii)$ of Theorem \ref{Thm1}, there must hold
$c_\epsilon\rightarrow +\infty$ as $\epsilon\rightarrow 0$. With no loss of generality we assume
$x_\epsilon\rightarrow x_0$ as $\epsilon\rightarrow 0$.
By definition, we have for any $u\in\mathscr{H}_{\textbf{G}}$
$$J_{8\pi\ell(1-\epsilon)}(u)=\frac{1}{2}\int_\Sigma|\nabla_gu|^2dv_g-8\pi\ell(1-\epsilon)\log
\int_\Sigma he^{u}dv_g.$$
It follows that for any $u\in\mathscr{H}_{\textbf{G}}$,
$J_{8\pi\ell}(u)=\lim_{\epsilon\rightarrow 0}J_{8\pi\ell(1-\epsilon)}(u)\geq \lim_{\epsilon\rightarrow 0}J_{8\pi\ell(1-\epsilon)}(u_\epsilon)$.
To prove $(ii)$ of Theorem \ref{Thm1}, it then suffices to prove
\begin{equation}\label{low-bdd}\lim_{\epsilon\rightarrow 0}J_{8\pi\ell(1-\epsilon)}(u_\epsilon)\geq -4\pi\ell\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)-8\pi\ell,\end{equation}
where $\widetilde{A}_x$ is defined as in  (\ref{G-repres}).
Let \begin{equation}\label{scal}r_\epsilon=\frac{\sqrt{\lambda_\epsilon}}{\sqrt{8\pi\ell(1-\epsilon)h(x_0)}}e^{-c_\epsilon/2}\end{equation}
and define a sequence of blow-up functions
\begin{equation}\label{varphi}\varphi_\epsilon(y)={u}_\epsilon\le(\exp_{x_\epsilon}(r_\epsilon y)\right)-c_\epsilon,\end{equation}
where $\exp_{x_\epsilon}:T_{x_\epsilon}\Sigma\,(\cong\mathbb{R}^2)\rightarrow \Sigma$ is an exponential map.
A straightforward calculation shows on the Euclidean ball $\mathbb{B}_{\delta r_\epsilon^{-1}}(0)$,
\begin{eqnarray}\label{phi-equa}
\Delta_{g_\epsilon}\varphi_\epsilon(y)=\frac{h(\exp_{x_\epsilon}(r_\epsilon y))}{h(x_0)}e^{\varphi_\epsilon(y)}
-\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}r_\epsilon^2.
\end{eqnarray}
Similar to (\cite{Yang-Zhu-SCM}, Lemma 2.7), we have the following:
\begin{lemma}\label{te-0}
Let $c_\epsilon$ and $r_\epsilon$ be as above. Then there exists some constant $C$ such that $r_\epsilon^2\leq Ce^{-\frac{1}{2}c_\epsilon}$.
\end{lemma}
\proof Testing the equation (\ref{E-L}) by $u_\epsilon$, one gets
$$\|\nabla_gu_\epsilon\|_{L^2(\Sigma,g)}^2=\frac{8\pi\ell(1-\epsilon)}{\lambda_\epsilon}\int_\Sigma hu_\epsilon
e^{u_\epsilon}dv_g\leq 8\pi\ell c_\epsilon.$$
This together with Chen's inequality (\ref{Chen}) leads to
$$\int_\Sigma he^{u_\epsilon}dv_g\leq C\int_\Sigma e^{4\pi\ell\frac{u_\epsilon^2}{\|\nabla_gu_\epsilon\|_{L^2(\Sigma,g)}^2}+
\frac{\|\nabla_gu_\epsilon\|_{L^2(\Sigma,g)}^2}{16\pi\ell}}dv_g\leq Ce^{
\frac{\|\nabla_gu_\epsilon\|_{L^2(\Sigma,g)}^2}{16\pi\ell}}\leq Ce^{\frac{1}{2}c_\epsilon}.$$
Hence we have by (\ref{scal}) that
$$r_\epsilon^2=\frac{e^{-c_\epsilon}}{8\pi\ell(1-\epsilon)h(x_0)}\int_\Sigma he^{u_\epsilon}dv_g\leq Ce^{-\frac{1}{2}c_\epsilon}.$$
This gives the desired result. $\hfill\Box$\\

The power of the above lemma is evident. It leads to $r_\epsilon\rightarrow 0$ and a stronger estimate $r_\epsilon c_\epsilon^q\rightarrow 0$ for
all $q>1$ as $\epsilon\rightarrow 0$, which is very important during the process of blow-up analysis.
Then applying elliptic estimates to (\ref{E-L}), we obtain
\begin{equation}\label{b-b}\varphi_\epsilon\rightarrow \varphi\,\,\, {\rm in}\,\,\, C^1_{\rm loc}(\mathbb{R}^2)\,\,\,{\rm as}\,\,\, \epsilon\rightarrow 0,\end{equation} where
$\varphi$ is a distributional solution of
$$\le\{\begin{array}{lll}
-\Delta_{\mathbb{R}^2}\varphi(y)=e^{\varphi(y)}\quad {\rm in}\quad\mathbb{R}^2\\[1.5ex]
\int_{\mathbb{R}^2}e^{\varphi(y)}dy<\infty.\end{array}
\right.$$
A classification theorem of Chen-Li \cite{C-L} gives
$\varphi(y)=-2\log(1+|y|^2/8)$ and thus
\begin{equation}\label{bubble-energy}\int_{\mathbb{R}^2}e^{\varphi(y)}dy=8\pi.\end{equation}

We now claim that
\begin{equation}\label{concentration}\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lambda_\epsilon^{-1}\int_{B_{Rr_\epsilon}(x_\epsilon)}he^{u_\epsilon}dv_g=\frac{1}{\ell}.\end{equation}
In fact, by a change of variable, (\ref{scal}) and (\ref{b-b})
\begin{eqnarray*}
\lambda_\epsilon^{-1}\int_{B_{Rr_\epsilon}(x_\epsilon)}he^{u_\epsilon}dv_g&=&(1+o_\epsilon(1))\lambda_\epsilon^{-1}
\int_{\mathbb{B}_R(0)}h(\exp_{x_\epsilon}(r_\epsilon y))e^{u_\epsilon(\exp_{x_\epsilon}(r_\epsilon y))}r_\epsilon^2dy\\
&=&\le(\frac{1}{8\pi\ell}+o_\epsilon(1)\right)\int_{\mathbb{B}_R(0)}e^{\varphi_\epsilon(y)}dy\\
&=&\le(\frac{1}{8\pi\ell}+o_\epsilon(1)\right)\le(\int_{\mathbb{B}_R(0)}e^{\varphi(y)} dy+o_\epsilon(1)\right).
\end{eqnarray*}
This together with (\ref{bubble-energy}) leads to (\ref{concentration}).

Recalling $\ell=\inf_{x\in\Sigma}I(x)=\inf_{x\in\Sigma}\sharp\textbf{G}(x)$, we now calculate $I(x_0)=\sharp\textbf{G}(x_0)$ as below.
\begin{lemma}\label{con-ell}
$I(x_0)=\ell$.
\end{lemma}
\proof Denote $I(x_0)=k_0$. Suppose that $k_0>\ell$. With no loss of generality we assume $\textbf{G}(x_0)=\{\sigma_1(x_0),\cdots,\sigma_{k_0}(x_0)\}$.
Fix some $\delta>0$ sufficiently small such that
$$B_{\delta}(\sigma_i(x_0))\cap B_{\delta}(\sigma_j(x_0))=\varnothing,\,\,\,\forall\,1\leq i<j\leq k_0.$$
Note that $x_\epsilon\rightarrow x_0$ and $r_\epsilon\rightarrow 0$ as $\epsilon\rightarrow 0$. For any fixed $R>0$, one gets $B_{Rr_\epsilon}(\sigma_i(x_\epsilon))\subset B_{\delta}(\sigma_i(x_0))$ for sufficiently small $\epsilon>0$.
Since
$h$ is $\textbf{G}$-invariant, we have by (\ref{concentration}) that
\begin{equation}\label{con-i}\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}{\lambda_\epsilon^{-1}}\int_{B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}he^{u_\epsilon}dv_g=\frac{1}{\ell},
\,\,\,\forall\, i=1,\cdots,k_0.\end{equation}
By (\ref{con-i}) and the assumption $k_0>\ell$, we obtain
$$1=\lambda_\epsilon^{-1}\int_\Sigma he^{u_\epsilon}dv_g\geq \sum_{i=1}^{k_0}\lambda_\epsilon^{-1}\int_{B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}
he^{u_\epsilon}dv_g=\frac{k_0}{\ell}>1,$$
which is impossible. Therefore $k_0=\ell$. $\hfill\Box$\\

Hereafter, by Lemma \ref{con-ell}, we assume $\sigma_i(x_0)\not=\sigma_j(x_0)$ for all $1\leq i<j\leq \ell$ and
$\textbf{G}(x_0)=\{\sigma_1(x_0),\cdots,\sigma_\ell(x_0)\}$. It follows from (\ref{con-i}) and the definition of $\lambda_\epsilon$ that
\begin{equation}\label{t-0}\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lambda_\epsilon^{-1}\int_{\Sigma\setminus \cup_{i=1}^\ell
B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}he^{u_\epsilon}dv_g=0.\end{equation}
Since the functions on the righthand side of (\ref{E-L}) are bounded in $L^1(\Sigma,g)$, we have by (\cite{Yang-Zhu-SCM}, Lemma 2.10) that
$u_\epsilon$ is bounded in $W^{1,q}(\Sigma,g)$ for any $1<q<2$. As a consequence there exists some $\widetilde{G}_{x_0}\in W^{1,q}(\Sigma,g)$
such that for $0<r<2q/(2-q)$,
$u_\epsilon$ converges to $\widetilde{G}_{x_0}$ weakly in  $W^{1,q}(\Sigma)$, strongly in $L^r(\Sigma)$ and almost everywhere in $\Sigma$.
In view of (\ref{con-i}) and (\ref{t-0}), $\widetilde{G}_{x_0}$ satisfies
$$\Delta_g\widetilde{G}_{x_0}=8\pi\ell\le(\frac{1}{\ell}\sum_{i=1}^{\ell}\delta_{\sigma_i(x_0)}-\frac{1}{{\rm Vol}_g(\Sigma)}\right)$$
in the distributional sense. Moreover, $\int_\Sigma \widetilde{G}_{x_0}dv_g=0$.
 Let $\widetilde{G}_{x_\epsilon}$ be a distributional solution of
 \begin{align*}
 \begin{cases}
      &\Delta_g\widetilde{G}_{x_\epsilon}=8\pi\le(\sum_{i=1}^{\ell}\delta_{\sigma_i(x_\epsilon)}-\frac{\ell}{{\rm Vol}_g(\Sigma)}\right) , \\
      & \int_\Sigma \widetilde{G}_{x_\epsilon}dv_g=0.
\end{cases}
\end{align*}
 The existence of  $\widetilde{G}_{x_\epsilon}$ is based on Proposition \ref{prop-2}.
 By elliptic estimates and Proposition \ref{prop-3}, $\widetilde{G}_{x_\epsilon}$ can be written as
 \begin{equation}\label{G-rep}\widetilde{G}_{x_\epsilon}(x)=-4\log r+\widetilde{A}_{x_\epsilon}+O(r)\end{equation}
 near $\sigma_i(x_\epsilon)$,  where $r$ denotes the geodesic distance between
 $x$ and $\sigma_i(x_\epsilon)$, $i=1,\cdots, \ell$. Similarly $\widetilde{G}_{x_0}$ can be represented by
 $\widetilde{G}_{x_0}(x)=-4\log r+\widetilde{A}_{x_0}+O(r)$ near $\sigma_i(x_0)$,  where $r$ denotes the geodesic distance between
 $x$ and $\sigma_i(x_0)$, $i=1,\cdots,\ell$. We now claim that
   \begin{equation}\label{c-v}\widetilde{A}_{x_\epsilon}\rightarrow \widetilde{A}_{x_0}\quad {\rm as}\quad \epsilon\rightarrow 0.\end{equation}
 To see this, with no loss of generality, we assume that $\sigma_1=Id$, the identity map. Write $A_{x_\epsilon}=\lim_{r\rightarrow 0}
 (G_{x_\epsilon}(x)+4\log r)$ and $A_{x_0}=\lim_{r\rightarrow 0}
 (G_{x_0}(x)+4\log r)$. By Proposition \ref{prop-2},
 $\widetilde{A}_{x_\epsilon}=A_{x_\epsilon}+\sum_{i=2}^\ell G(\sigma_i(x_\epsilon),x_\epsilon)$.
 Since $A_{x_\epsilon}\rightarrow A_{x_0}$ and $x_\epsilon\rightarrow x_0$ as $\epsilon\rightarrow 0$, we immediately have
 $\widetilde{A}_{x_\epsilon}\rightarrow \widetilde{A}_{x_0}=A_{x_0}+\sum_{i=2}^\ell G(\sigma_i(x_0),x_0)$ as
 $\epsilon\rightarrow 0$. This confirms our claim (\ref{c-v}).

 One can easily see that
 $$\Delta_g(u_\epsilon-\widetilde{G}_{x_\epsilon})\geq 0\quad{\rm in}\quad \Sigma\setminus\cup_{i=1}^\ell B_{Rr_\epsilon}(\sigma_i(x_\epsilon)).$$
 In view of (\ref{b-b}), (\ref{G-rep}) and (\ref{c-v}), for all $1\leq i\leq \ell$, there holds
 $$(u_\epsilon-\widetilde{G}_{x_\epsilon})|_{\partial B_{Rr_\epsilon}(x_\epsilon)}=-c_\epsilon+2\log\lambda_\epsilon-\widetilde{A}_{x_0}-
 2\log(\pi\ell h(x_0))+o_\epsilon(1)+o_R(1).$$
 By the maximum principle, we have on $\Sigma\setminus\cup_{i=1}^\ell B_{Rr_\epsilon}(\sigma_i(x_\epsilon))$
 \begin{equation}\label{max-prin}u_\epsilon-\widetilde{G}_{x_\epsilon}\geq-c_\epsilon+2\log\lambda_\epsilon-\widetilde{A}_{x_0}-
 2\log(\pi\ell h(x_0))+o_\epsilon(1)+o_R(1).\end{equation}
 By (\ref{b-b}), one calculates
 \begin{eqnarray}\nonumber
 \int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}|\nabla_gu_\epsilon|^2dv_g&=&\ell\int_{B_{Rr_\epsilon}(x_\epsilon)}
 |\nabla_gu_\epsilon|^2dv_g\\\nonumber
 &=&\ell\le(\int_{\mathbb{B}_R(0)}|\nabla \varphi|^2dx+o_\epsilon(1)\right)\\\label{en-iner}
 &=&16\pi\ell\log\le(1+\frac{R^2}{8}\right)-16\pi\ell+o_\epsilon(1)+o_R(1),
 \end{eqnarray}
 where $o_\epsilon(1)\rightarrow 0$ as $\epsilon\rightarrow 0$ for any fixed $R>0$, $o_R(1)\rightarrow 0$ as $R\rightarrow \infty$.

 It follows from the divergence theorem and (\ref{E-L}) that
 \begin{eqnarray}\nonumber
 \int_{\Sigma_{R,\epsilon}}|\nabla_gu_\epsilon|^2dv_g&=&
 \int_{\Sigma_{R,\epsilon}}u_\epsilon\Delta_gu_\epsilon dv_g+
 \int_{\partial \Sigma_{R,\epsilon}}u_\epsilon\frac{\partial u_\epsilon}{\partial \nu}ds_g\\\nonumber
 &=&8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}\lambda_\epsilon^{-1}hu_\epsilon e^{u_\epsilon}dv_g
 -\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\int_{\Sigma_{R,\epsilon}} u_\epsilon dv_g\\
 &&\quad+
 \int_{\partial \Sigma_{R,\epsilon}}u_\epsilon\frac{\partial u_\epsilon}{\partial \nu}ds_g,\label{a-0}
 \end{eqnarray}
  where  $\nu$ defined on the boundary $\partial\Omega$ denotes the outer unit vector with respect to the domain $\Omega$ and we write for simplicity
 $$\Sigma_{R,\epsilon}=\Sigma\setminus\cup_{i=1}^\ell B_{Rr_\epsilon}(\sigma_i(x_\epsilon)).$$
 By (\ref{max-prin}),
 \begin{eqnarray}
 &&8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}\lambda_\epsilon^{-1}hu_\epsilon e^{u_\epsilon}dv_g\nonumber\\&&\geq
 8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}\widetilde{G}_{x_\epsilon}\frac{h e^{u_\epsilon}}{\lambda_\epsilon}dv_g
 -8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}(c_\epsilon-2\log\lambda_\epsilon)\frac{h e^{u_\epsilon}}{\lambda_\epsilon}dv_g\nonumber\\
 &&\quad-8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}\le(\widetilde{A}_{x_0}+2\log(\pi\ell h(x_0))+o_\epsilon(1)+o_R(1)\right)
 \frac{h e^{u_\epsilon}}{\lambda_\epsilon}dv_g.\label{a-1}
 \end{eqnarray}
 By the divergence and (\ref{E-L}),
 \begin{eqnarray}\nonumber
 8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}\frac{\widetilde{G}_{x_\epsilon}h e^{u_\epsilon}}{\lambda_\epsilon}dv_g
 &=&\int_{\Sigma_{R,\epsilon}}\widetilde{G}_{x_\epsilon}\le(\Delta_gu_\epsilon+\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\right)dv_g\\\nonumber
 &=&\int_{\Sigma_{R,\epsilon}}u_\epsilon\Delta_g\widetilde{G}_{x_\epsilon}dv_g-\int_{\partial \Sigma_{R,\epsilon}}\widetilde{G}_{x_\epsilon}
 \frac{\partial u_\epsilon}{\partial \nu}ds_g\\\nonumber
 &&+\int_{\partial\Sigma_{R,\epsilon}}u_\epsilon\frac{\partial \widetilde{G}_{x_\epsilon}}{\partial\nu}ds_g-\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}
 \int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}\widetilde{G}_{x_\epsilon}dv_g\\\nonumber
 &=&-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\Sigma_{R,\epsilon}}u_\epsilon dv_g-
 \int_{\partial\Sigma_{R,\epsilon}}\widetilde{G}_{x_\epsilon}\frac{\partial u_\epsilon}{\partial\nu}ds_g\\
 &&+\int_{\partial \Sigma_{R,\epsilon}}u_\epsilon\frac{\partial \widetilde{G}_{x_\epsilon}}{\partial\nu}ds_g
 -\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\int_{\Sigma\setminus\Sigma_{R,\epsilon}}\widetilde{G}_{x_\epsilon}
 dv_g.
 \end{eqnarray}
 Also one has
 \begin{eqnarray}
 \nonumber 8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}(c_\epsilon-2\log\lambda_\epsilon)\frac{he^{u_\epsilon}}{\lambda_\epsilon}dv_g&=&
 (c_\epsilon-2\log\lambda_\epsilon)\int_{\Sigma_{R,\epsilon}}\le(\Delta_gu_\epsilon+\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\right) dv_g\\
 \nonumber&=&(c_\epsilon-2\log\lambda_\epsilon)\le\{\int_{\cup_{i=1}^\ell\partial B_{Rr_\epsilon(\sigma_i(x_\epsilon))}}\frac{\partial u_\epsilon}{\partial \nu}ds_g\right.\\
  &&\le.+\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\le({\rm Vol}_g(\Sigma)-\ell{{\rm Vol}_g(B_{Rr_\epsilon}(x_\epsilon))}\right)\right\}
 \end{eqnarray}
 and in view of (\ref{t-0}),
 \begin{equation}\label{o-1}
 8\pi\ell(1-\epsilon)\int_{\Sigma_{R,\epsilon}}(\widetilde{A}_{x_0}+2\log(\pi\ell h(x_0))+o_\epsilon(1)+o_R(1))\frac{he^{u_\epsilon}}{\lambda_\epsilon}dv_g
 =o_\epsilon(1)+o_R(1).
 \end{equation}
 Inserting (\ref{a-1})-(\ref{o-1}) into (\ref{a-0}), we have
 \begin{eqnarray}\nonumber
 \int_{\Sigma_{R,\epsilon}}|\nabla_gu_\epsilon|^2dv_g&\geq&\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(x_\epsilon)}u_\epsilon dv_g
 -\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(x_\epsilon)}\widetilde{G}_{x_\epsilon}dv_g\\\nonumber
 &&-\int_{\cup_{i=1}^\ell\partial B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}\frac{\partial u_\epsilon}{\partial \nu}(u_\epsilon-\widetilde{G}_{x_\epsilon}+c_\epsilon-2\log\lambda_\epsilon)ds_g\\\nonumber
 &&-\int_{\cup_{i=1}^\ell \partial B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}u_\epsilon\frac{\partial\widetilde{G}_{x_\epsilon}}{\partial\nu}ds_g+
 \frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}\int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(x_\epsilon)}u_\epsilon dv_g\\\nonumber
 &&+\frac{8\pi\ell(1-\epsilon)}{{\rm Vol}_g(\Sigma)}(-c_\epsilon+2\log\lambda_\epsilon)\le({\rm Vol}_g(\Sigma)-\ell{{\rm Vol}_g(B_{Rr_\epsilon}(x_\epsilon))}\right)\\\label{geq-1}
 &&+o_\epsilon(1)+o_R(1).
 \end{eqnarray}
 In view of (\ref{varphi}) and (\ref{b-b}), for $y\in\partial \mathbb{B}_{R}(0)$, there holds
 \begin{eqnarray*}
 \frac{\partial u_\epsilon}{\partial\nu}(\exp_{x_\epsilon}(r_\epsilon y))
 =\frac{1}{r_\epsilon}\le(\frac{\partial \varphi}{\partial r}(y)+o_\epsilon(1)\right)
 =\frac{1}{r_\epsilon}\le(-\frac{R/2}{1+R^2/8}+o_\epsilon(1)\right).
 \end{eqnarray*}
 It follows that for $i=1,\cdots,\ell$
 $$\le.-\frac{\partial u_\epsilon}{\partial \nu}\right|_{\partial B_{Rr\epsilon}(\sigma_i(x_\epsilon))}=\frac{1}{r_\epsilon}\le(\frac{R/2}{1+R^2/8}+o_\epsilon(1)\right).$$
 This together with (\ref{max-prin}) leads to
 \begin{eqnarray}\nonumber
 &&-\int_{\cup_{i=1}^\ell\partial B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}\frac{\partial u_\epsilon}{\partial \nu}(u_\epsilon-\widetilde{G}_{x_\epsilon}+c_\epsilon-2\log\lambda_\epsilon)ds_g\\\label{b-0}
 &&\geq \frac{\pi\ell R^2}{1+R^2/8}(-2\log(\pi\ell h(x_0))-\widetilde{A}_{x_0})+o_\epsilon(1)+o_R(1).
 \end{eqnarray}
 By (\ref{b-b}) and (\ref{G-rep}), we have
 \begin{equation}\label{a-2}-\int_{\cup_{i=1}^\ell\partial B_{Rr_\epsilon}(\sigma_i(x_\epsilon))}u_\epsilon\frac{\partial\widetilde{G}_{x_\epsilon}}{\partial\nu}ds_g=
 8\pi\ell c_\epsilon-16\pi\ell \log\le(1+\frac{R^2}{8}\right)+o_\epsilon(1)+o_R(1).\end{equation}
 It is easy to see that
 \begin{equation}\label{a-3}\int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(x_\epsilon)}\widetilde{G}_{x_\epsilon}dv_g=o_\epsilon(1),\quad
 \int_{\cup_{i=1}^\ell B_{Rr_\epsilon}(x_\epsilon)}u_\epsilon dv_g=o_\epsilon(1)\end{equation}
 and \begin{equation}\label{a-4}(-c_\epsilon+2\log\lambda_\epsilon){\rm Vol}_g(B_{Rr_\epsilon}(\sigma_i(x_\epsilon)))=o_\epsilon(1).\end{equation}
 Inserting (\ref{b-0})-(\ref{a-4}) into (\ref{geq-1}), we have
 \begin{eqnarray*}
 \int_{\Sigma_{R,\epsilon}}|\nabla_gu_\epsilon|^2dv_g&\geq&8\pi\ell\le(-2\log(\pi\ell h(x_0))-\widetilde{A}_{x_0}\right)
 +16\pi\ell(1-\epsilon)\log\lambda_\epsilon\\
 &&-16\pi\ell\log\le(1+\frac{R^2}{8}\right)+o_\epsilon(1)+o_R(1).
 \end{eqnarray*}
 This together with (\ref{en-iner}) leads to
 \begin{eqnarray}\nonumber
 \int_{\Sigma}|\nabla_gu_\epsilon|^2dv_g&\geq&8\pi\ell\le(-2\log(\pi\ell h(x_0))-\widetilde{A}_{x_0}\right)
 +16\pi\ell(1-\epsilon)\log\lambda_\epsilon\\\label{g-e}
 &&-16\pi\ell+o_\epsilon(1)+o_R(1).
 \end{eqnarray}
 In view of (\ref{E-L}), it follows from (\ref{g-e}) that
 \begin{eqnarray*}
 J_{8\pi\ell(1-\epsilon)}(u_\epsilon)&=&\frac{1}{2}\int_\Sigma|\nabla_gu_\epsilon|^2dv_g-8\pi\ell(1-\epsilon)\log\int_\Sigma he^{u_\epsilon}dv_g\\
 &\geq&4\pi\ell\le(-2\log(\pi\ell h(x_0))-\widetilde{A}_{x_0}\right)
 -8\pi\ell+o_\epsilon(1)+o_R(1).
 \end{eqnarray*}
 Therefore
 $$\lim_{\epsilon\rightarrow 0}J_{8\pi\ell(1-\epsilon)}(u_\epsilon)\geq 4\pi\ell\le(-2\log(\pi\ell h(x_0))-\widetilde{A}_{x_0}\right)
 -8\pi\ell,$$
 which immediately leads to (\ref{low-bdd}) and completes the proof of $(ii)$ of Theorem \ref{Thm1}. $\hfill\Box$

\subsection{Proof of Theorem \ref{Thm2}}
Suppose that (\ref{meanfield}) has no solution. By Theorem \ref{Thm1},
$$\inf_{u\in\mathscr{H}_{\textbf{G}}}J_{8\pi\ell}(u)\geq -4\pi\ell\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)-8\pi\ell,$$
where $\widetilde{A}_x$ is defined as in  (\ref{G-repres}). Hence
$$J_{8\pi\ell}(0)=-8\pi\ell\log\int_\Sigma hdv_g\geq -4\pi\ell\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)-8\pi\ell,$$
which contradicts (\ref{cond}). Therefore (\ref{meanfield}) has a solution. $\hfill\Box$

\section{Proof of Theorem \ref{Thm3}}\label{Sec5}
In this section, under the assumptions of Theorem \ref{Thm3}, we shall construct test functions to show
\begin{equation}\label{test}\inf_{u\in\mathscr{H}_{\textbf{G}}}J_{8\pi\ell}(u)< -4\pi\ell\max_{x\in\Sigma}
(2\log(\pi\ell h(x))+\widetilde{A}_x)-8\pi\ell,\end{equation}
where $\widetilde{A}_x$ is defined as in  (\ref{G-repres}). This together with Theorem \ref{Thm1} concludes
Theorem \ref{Thm3}. In the sequel,
we assume $\textbf{G}=\{\sigma_1,\cdots,\sigma_\ell\}$,
$I(x)=\sharp\textbf{G}(x)\equiv \ell$ for all $x\in\Sigma$.
Pick up some point $p\in \Sigma$ such that
\begin{equation}\label{ma-x}\widetilde{A}_p+2\log(\pi\ell h(p))=\max_{x\in\Sigma}(\widetilde{A}_x+2\log(\pi\ell h(x))).\end{equation}
It follows from $I(p)=\ell$ that  $\sigma_1(p),\cdots,\sigma_\ell(p)$ are different points on $\Sigma$.
Let \begin{equation}\label{delta}\delta=\frac{1}{4}\min\le\{{\rm inj}_g(\Sigma),\min_{1\leq i<j\leq \ell}d_g(\sigma_i(p),\sigma_j(p))\right\}.\end{equation}
Choose a normal coordinate system $(B_\delta(p),\exp_{p}^{-1};\{y^1,y^2\})$ near $p$.
More precisely, fixing an orthonormal basis $\{e_1,e_2\}$ of the tangent space $T_p\Sigma$, one can
write $y=(y^1,y^2)=\exp_p^{-1}(x)$ if the exponential map  $\exp_p: T_p\Sigma(\cong\mathbb{R}^2)\rightarrow\Sigma$ maps
the tangent vector $y^1e_1+y^2e_2$ to
the point $x\in B_\delta(p)$.  One also denotes $\exp_p(y^1e_1+y^2e_2)$ by $\exp_p(y)$, and thus
$\exp_p^{-1}(B_\delta(p))=\mathbb{B}_\delta(0)$.
Let $\widetilde{G}_p$ be a Green function satisfying
\begin{equation}\label{G-tilde}
\le\{\begin{array}{lll}
\Delta_g\widetilde{G}_p=8\pi\sum_{i=1}^\ell\delta_{\sigma_i(p)}-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\\[1.5ex]
\int_\Sigma \widetilde{G}_pdv_g=0.
\end{array}\right.
\end{equation}
The existence of such a $\widetilde{G}_p$ is based on Proposition \ref{prop-1}. By Propositions \ref{prop-2} and
\ref{prop-3},  we have
 \begin{equation}\label{G-inv}\widetilde{G}_p(\sigma_i(y))=\widetilde{G}_p(y),\quad i=1,\cdots,\ell.\end{equation}
In the above mentioned normal coordinate system $(B_\delta(p),\exp_{p}^{-1};\{y^1,y^2\})$, by elliptic estimates,
 $\widetilde{G}_p$ can be written as
\begin{equation}\label{Green-repres}
\widetilde{G}_p(\exp_p(y))=-4\log r+\widetilde{A}_p+\alpha(y)+\beta(y),
\end{equation}
where $r=|y|=d_g(p,\exp_p(y))$, $\widetilde{A}_p$ is a constant, $\alpha(y)=b_1y^1+b_2y^2$, $\beta(y)\in C^1(\mathbb{B}_\delta(0))$ and
$$\beta(y)=c_1(y^1)^2+2c_2y^1y^2+c_3(y^2)^2+O(r^3).$$
To proceed, we have the following:
\begin{lemma}\label{c1-3-k} Let $c_1$ and $c_3$ are constants in (\ref{Green-repres}),
$K(p)$ be the Gaussian curvature of $(\Sigma,g)$ at $p$.  Then the following identity holds
$$c_1+c_3+\frac{2}{3}K(p)=\frac{4\pi\ell}{{\rm Vol}_g(\Sigma)}.$$
\end{lemma}

\proof We modify the argument of (\cite{DJLW}, Proposition 3.2). In a normal coordinate system near $p$,  the Riemannian metric
can be written as $g=dr^2+g^2(r,\theta)d\theta^2$ with
\begin{equation}\label{g-rtheta}g(r,\theta)=r-\frac{K(p)}{6}r^3+\phi(\cos\theta,\sin\theta)r^4+O(r^5),\end{equation}
where $\phi(s,t)=\sum_{j=0}^3a_js^{3-j}t^j$ is a third order homogenous polynomial.

By the divergence theorem, we have for any $0<r<\delta$
\begin{equation}\label{div}\int_{\cup_{i=1}^\ell\partial B_r(\sigma_i(p))}\frac{\partial \widetilde{G}_p}{\partial n}ds_g=\int_{\Sigma\setminus \cup_{i=1}^\ell B_r(\sigma_i(p))}
\Delta_g\widetilde{G}_p dv_g=-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\Sigma\setminus \cup_{i=1}^\ell B_r(\sigma_i(p))}dv_g.\end{equation}
Note that $\sigma^\ast g(x)=g(\sigma(x))$ for all $\sigma\in \textbf{G}$ and $x\in\Sigma$. In view of (\ref{G-inv}), (\ref{Green-repres}) and (\ref{g-rtheta}), one calculates
\begin{eqnarray*}
\int_{\cup_{i=1}^\ell\partial B_r(\sigma_i(p))}\frac{\partial \widetilde{G}_p}{\partial n}ds_g
&=&\ell\int_0^{2\pi}\le(-\frac{4}{r}+2c_1r\cos^2\theta+4c_2r\cos\theta\sin\theta+2c_3r\sin^2\theta+O(r^2)\right)\\
&&\quad\quad\quad\le(r-\frac{K(p)}{6}r^3+O(r^4)\right)d\theta\\
&=& -8\pi\ell+2\pi\ell\le(c_1+c_3+\frac{2}{3}K(p)\right)r^2+O(r^3)
\end{eqnarray*}
and
$$-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\Sigma\setminus \cup_{i=1}^\ell B_r(\sigma_i(p))}dv_g=-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}
\le({\rm Vol}_g(\Sigma)-\pi\ell r^2+O(r^4)\right).$$
Inserting the above two estimates to (\ref{div}) and comparing the the terms involving $r^2$, we get the desired result. $\hfill\Box$\\

We define a sequence of functions $(\phi_\epsilon)_{\epsilon>0}$ by
$$\phi_\epsilon(x)=\le\{
\begin{array}{lll}
c-2\log\le(1+\frac{r^2}{8\epsilon^2}\right)+\widetilde{A}_p+\alpha(\exp_p^{-1}(\sigma_i^{-1}(x))),&x\in B_{R\epsilon}(\sigma_i(p)),\,\, i=1,\cdots,\ell\\[1.5ex]
\widetilde{G}_p(x)-\eta(\sigma_i^{-1}(x))\,\beta(\exp_p^{-1}(\sigma_i^{-1}(x))),&x\in B_{2R\epsilon}(\sigma_i(p))\setminus B_{R\epsilon}(\sigma_i(p))\\[1.5ex]
\widetilde{G}_p(x),&x\in\Sigma\setminus \cup_{i=1}^\ell B_{2R\epsilon}(\sigma_i(p)),
\end{array}\right.$$
where $R\epsilon\rightarrow 0$ as $\epsilon\rightarrow 0$, $R$ and $c$ are constants depending only on $\epsilon$ and will be determined later, $r=r(x)$ denotes the geodesic distance between $x$ and $\sigma_i(p)$ for $x\in B_{R\epsilon}(\sigma_i(p))$,
$\eta\in C_0^\infty(B_{2R\epsilon}(p))$  satisfies $0\leq \eta\leq 1$, $\eta\equiv 1$ on $B_{R\epsilon}(p)$ and $|\nabla_g\eta|\leq 4/(R\epsilon)$,
both $\alpha$ and $\beta$ are functions defined as in (\ref{Green-repres}).

To ensure $\phi_\epsilon\in W^{1,2}(\Sigma,g)$, we require for all $x\in \partial B_{R\epsilon}(\sigma_i(p))$, there holds
$$c-2\log\le(1+\frac{r^2}{8\epsilon^2}\right)+\widetilde{A}_p+\alpha(\exp_p^{-1}(\sigma_i^{-1}(x)))=\widetilde{G}_p(x)-\eta(\sigma_i(x)\,\beta(\exp_p^{-1}(\sigma_i^{-1}(x))).$$
This together with (\ref{Green-repres}) implies that
\begin{equation}\label{c}c=2\log\le(1+\frac{R^2}{8}\right)-4\log (R\epsilon).\end{equation}
One can easily check that
$$\phi_\epsilon(\sigma_i(x))=\phi_\epsilon(x),\quad\forall x\in\Sigma,\,\,\forall i=1,\cdots,\ell.$$
A straightforward calculation shows
\begin{eqnarray}\nonumber
\int_{B_{R\epsilon}(\sigma_i(p))}|\nabla_g\phi_\epsilon|^2dv_g&=&\int_{B_{R\epsilon}(p)}|\nabla_g\phi_\epsilon|^2dv_g\\
\nonumber&=&\int_{B_{R\epsilon}(p)}\frac{16r^2}{(r^2+8\epsilon^2)^2}dv_g+\int_{B_{R\epsilon}(p)}|\nabla_g(\alpha(\exp_p^{-1}(x)))|^2dv_g\\
&&\quad-
\int_{B_{R\epsilon}(p)}\frac{8r\nabla_gr\nabla_g(\alpha(\exp_p^{-1}(x)))}{r^2+8\epsilon^2}dv_g.\label{energ-re}
\end{eqnarray}
Since $\int_0^{2\pi}\phi(\cos\theta,\sin\theta)d\theta=0$, we have
\begin{eqnarray*}
\int_{B_{R\epsilon}(p)}\frac{16r^2}{(r^2+8\epsilon^2)^2}dv_g&=&\int_0^{2\pi}d\theta\int_0^{R\epsilon}\frac{16r^2}{(r^2+8\epsilon^2)^2}
(r-\frac{K(p)r^3}{6}+\phi(\cos\theta,\sin\theta)r^4+O(r^5))dr\\
&=&16\pi\log\le(1+\frac{R^2}{8}\right)-\frac{16\pi R^2}{R^2+8}+\frac{128}{3}\pi K(p)\epsilon^2\log\le(1+\frac{R^2}{8}\right)\\&&-\frac{8}{3}\pi K(p)(R\epsilon)^2
+O(\epsilon^2)+O((R\epsilon)^4).
\end{eqnarray*}
In a normal coordinate system near $p$, $g^{ij}(y)=\delta^{ij}+O(r^2)$, there holds
\begin{eqnarray*}
\int_{B_{R\epsilon}(p)}|\nabla_g(\alpha(\exp_p^{-1}(x)))|^2dv_g&=&\int_{\mathbb{B}_{R\epsilon}(0)}
g^{ij}(y)\partial_i(b_1y^1+b_2y^2)\partial_j(b_1y^1+b_2y^2)(1+O(r^2))dy\\
&=& \pi(b_1^2+b_2^2)(R\epsilon)^2+O((R\epsilon)^4).
\end{eqnarray*}
Moreover,
\begin{eqnarray*}
\int_{B_{R\epsilon}(p)}\frac{8r\nabla_gr\nabla_g(\alpha(\exp_p^{-1}(x)))}{r^2+8\epsilon^2}dv_g&=&4\int_0^{2\pi}d\theta
\int_0^{R\epsilon}\frac{\partial_rr^2\partial_r(b_1r\cos\theta+b_2r\sin\theta)}{r^2+8\epsilon^2}g(r,\theta)dr\\
&=&O((R\epsilon)^4).
\end{eqnarray*}
Inserting the above three estimates to (\ref{energ-re}), we have
\begin{eqnarray}\nonumber
\int_{B_{R\epsilon}(\sigma_i(p))}|\nabla_g\phi_\epsilon|^2dv_g&=&16\pi\log\le(1+\frac{R^2}{8}\right)-\frac{16\pi R^2}{R^2+8}+\frac{128}{3}\pi K(p)\epsilon^2\log\le(1+\frac{R^2}{8}\right)\\&&-\frac{8}{3}\pi K(p)(R\epsilon)^2+\pi(b_1^2+b_2^2)(R\epsilon)^2\nonumber\\&&
+O((R\epsilon)^4)+O(\epsilon^2)+O((R\epsilon)^4\log(1+{R^2}/{8})).\label{Repsilon}
\end{eqnarray}

Write $p_i=\sigma_i(p)$, $\eta_i=\eta\circ (\exp_p^{-1}\sigma_i^{-1})$, $\alpha_i=\alpha\circ \sigma_i^{-1}$ and $\beta_i=\beta\circ \sigma_i^{-1}$.
By the divergence theorem and (\ref{G-tilde}), we calculate
\begin{eqnarray}\nonumber
\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}|\nabla_g\phi_\epsilon|^2dv_g&=&
\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}|\nabla_g\widetilde{G}_p|^2dv_g+
\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}|\nabla_g(\eta_i\beta_i)|^2dv_g\\\nonumber
&&-2\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\nabla_g\widetilde{G}_p\nabla_g(\eta_i\beta_i)dv_g\\\nonumber
&=&-\int_{\cup_{i=1}^\ell\partial B_{R\epsilon}(p_i)}\widetilde{G}_p\frac{\partial\widetilde{G}_p}{\partial n}ds_g+
\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}\widetilde{G}_p\Delta_g\widetilde{G}_p dv_g\\\nonumber
&&+\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}|\nabla_g(\eta_i\beta_i)|^2dv_g\\\label{out}
&&-2\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\eta_i\beta_i\Delta_g\widetilde{G}_p dv_g+
2\int_{\cup_{i=1}^\ell\partial B_{R\epsilon}(p_i)}\eta_i\beta_i\frac{\partial\widetilde{G}_p}{\partial n}ds_g.\quad
\end{eqnarray}
By (\ref{Green-repres}),  we have in a normal polar coordinate system near $p$
\begin{eqnarray}\nonumber \widetilde{G}_p(\exp_p(y))&=&-4\log r+\widetilde{A}_p+b_1r\cos\theta+b_2r\sin\theta+c_1r^2\cos^2\theta+2c_2r^2\cos\theta\sin\theta\\[1.2ex]
&&\quad+c_3r^2\sin^2\theta+\varrho(\cos\theta,\sin\theta)r^3+O(r^4),\label{G-polar}\end{eqnarray}
where $y=(y^1,y^2)=(r\cos\theta,r\sin\theta)$, $\varrho(s,t)$ is a third order homogenous polynomial with respect to $s$ and $t$. Obviously
\begin{align}\label{patial-G}
\frac{\partial\widetilde{G}_p}{\partial n}=&-\frac{4}{r}+b_1\cos\theta+b_2\sin\theta+2c_1r\cos^2\theta+4c_2r\cos\theta\sin\theta+2c_3r\sin^2\theta\nonumber\\
&+3r^2\varrho(\cos\theta,\sin\theta)+O(r^3).
\end{align}
Combining (\ref{g-rtheta}), (\ref{G-polar}) and (\ref{patial-G}), one has
\begin{eqnarray}\nonumber
-\int_{\cup_{i=1}^\ell\partial B_{R\epsilon}(p_i)}\widetilde{G}_p\frac{\partial\widetilde{G}_p}{\partial n}ds_g&=&-\ell\int_
{\partial B_{R\epsilon}(p)}\widetilde{G}_p\frac{\partial\widetilde{G}_p}{\partial n}ds_g\\\nonumber
&=&-32\pi\ell\log(R\epsilon)+8\pi\ell \widetilde{A}_p+4\pi\ell(c_1+c_3)(R\epsilon)^2\\\nonumber
&&-\pi\ell(b_1^2+b_2^2)(R\epsilon)^2-2\pi\ell(c_1+c_3+\frac{2}{3}K(p))\widetilde{A}_p(R\epsilon)^2\\\label{est-1}
&&+8\pi\ell(c_1+c_3+\frac{2}{3}K(p))(R\epsilon)^2\log(R\epsilon)+O((R\epsilon)^4\log(R\epsilon)).
\end{eqnarray}
Similarly
\begin{eqnarray}\nonumber
\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}\widetilde{G}_p\Delta_g\widetilde{G}_p dv_g&=&
-\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}\widetilde{G}_pdv_g
=\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\cup_{i=1}^\ell B_{R\epsilon}(p_i)}\widetilde{G}_pdv_g\\\nonumber
&=&-\frac{32\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}(R\epsilon)^2\log(R\epsilon)+\frac{16\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}(R\epsilon)^2
+\frac{8\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}\widetilde{A}_p(R\epsilon)^2\\&&\quad+O((R\epsilon)^4\log(R\epsilon)),\label{est-2}
\end{eqnarray}
\begin{equation}\label{est-3}\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}|\nabla_g(\eta_i\beta_i)|^2dv_g=O((R\epsilon)^4),\end{equation}
\begin{equation}\label{est-4}
 -2\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\eta_i\beta_i\Delta_g\widetilde{G}_p dv_g=
\frac{16\pi\ell}{{\rm Vol}_g(\Sigma)}\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\eta_i\beta_idv_g
=O((R\epsilon)^4)
\end{equation}
and
\begin{equation}\label{est-5}2\int_{\cup_{i=1}^\ell\partial B_{R\epsilon}(p_i)}\eta_i\beta_i\frac{\partial\widetilde{G}_p}{\partial n}ds_g=
-8\pi\ell(c_1+c_3)(R\epsilon)^2+O((R\epsilon)^4).\end{equation}
Inserting (\ref{est-1})-(\ref{est-5}) to (\ref{out}), we obtain
\begin{eqnarray*}
\int_{\Sigma\setminus\cup_{i=1}^\ell B_{R\epsilon}(p_i)}|\nabla_g\phi_\epsilon|^2dv_g&=&-32\pi\ell\log(R\epsilon)+8\pi\ell \widetilde{A}_p+
4\pi\ell(c_1+c_3)(R\epsilon)^2\\
&&-\pi\ell(b_1^2+b_2^2)(R\epsilon)^2-2\pi\ell(c_1+c_3+\frac{2}{3}K(p))\widetilde{A}_p(R\epsilon)^2\\&&+8\pi\ell(c_1+c_3+\frac{2}{3}K(p))(R\epsilon)^2\log(R\epsilon)\\
&&-\frac{32\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}(R\epsilon)^2\log(R\epsilon)+\frac{16\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}(R\epsilon)^2
+\frac{8\pi^2\ell^2}{{\rm Vol}_g(\Sigma)}\widetilde{A}_p(R\epsilon)^2\\
&&-8\pi\ell(c_1+c_3)(R\epsilon)^2+O((R\epsilon)^2\log(R\epsilon)).
\end{eqnarray*}
This together with (\ref{Repsilon}) and Lemma \ref{c1-3-k} leads to
\begin{eqnarray}\nonumber
\int_\Sigma|\nabla_g\phi_\epsilon|^2dv_g&=&16\pi\ell\log\le(1+\frac{R^2}{8}\right)-\frac{16\pi\ell R^2}{R^2+8}+\frac{128}{3}\pi\ell K(p)\epsilon^2
\log\le(1+\frac{R^2}{8}\right)\\&&-32\pi\ell\log(R\epsilon)+8\pi\ell \widetilde{A}_p+O((R\epsilon)^4\log(R\epsilon))+O(\epsilon^2)\nonumber\\
&&+O((R\epsilon)^4\log{1+\frac{R^2}{8}}).\label{energy}
\end{eqnarray}

Now we estimate the average of the integral of $\phi_\epsilon$ as follows.
\begin{eqnarray}\nonumber
\overline{\phi}_\epsilon&=&\frac{1}{{\rm Vol}_g(\Sigma)}\int_\Sigma \phi_\epsilon dv_g\\\nonumber
&=&\frac{1}{{\rm Vol}_g(\Sigma)}\le(\int_{\cup_{i=1}^\ell B_{R\epsilon}(p_i)}\le(c-2\log\le(1+\frac{r^2}{8\epsilon^2}\right)+\widetilde{A}_p+\alpha_i\right)dv_g-
\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\eta_i\beta_i dv_g\right.\\\nonumber
&&\quad\le.+\int_{\Sigma\setminus\cup_{i=1}^\ell B_{2R\epsilon}(p_i)}\widetilde{G}_pdv_g+
\int_{\cup_{i=1}^\ell B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}\widetilde{G}_pdv_g\right)\\\label{average}
&=&-\frac{16\pi\ell}{{\rm Vol}_g(\Sigma)}\epsilon^2\log\le(1+\frac{R^2}{8}\right)+O((R\epsilon)^4\log(R\epsilon))+O((R\epsilon)^4\log(1+R^2/8)).
\end{eqnarray}

Next we estimate $\int_\Sigma he^{\phi_\epsilon}dv_g$.
\begin{eqnarray}\nonumber
\int_{B_{R\epsilon}(p_i)}e^{\phi_\epsilon}dv_g&=&\int_{B_{R\epsilon}(p)}e^{\phi_\epsilon}dv_g\\\nonumber
&=& e^{c+\widetilde{A}_p}\int_0^{2\pi}d\theta\int_0^{R\epsilon}\frac{e^{b_1r\cos\theta+b_2r\sin\theta}}{(1+r^2/(8\epsilon^2))^2}
\le(r-\frac{K(p)}{6}r^3+O(r^4)\right)dr\\\nonumber
&=&\le(\frac{1}{8}+\frac{1}{R^2}\right)^2e^{\widetilde{A}_p}\epsilon^{-2}\le\{8\pi \frac{R^2}{R^2+8}-\frac{32}{3}\pi K(p)\epsilon^2\log\le(1+\frac{R^2}{8}\right)\right.\\
&&\qquad\qquad \le.+ 16\pi(b_1^2+b_2^2)\epsilon^2\log\le(1+\frac{R^2}{8}\right)+O(\epsilon^2)\right\}.\label{integ-1}
\end{eqnarray}
Let $\delta$ be defined as in (\ref{delta}).
Note that for $y\in \mathbb{B}_\delta(0)$, there holds
\begin{eqnarray*}
e^{\widetilde{G}_p(\exp_p(y))}&=&r^{-4}e^{\widetilde{A}_p}\le\{1+b_1r\cos\theta+b_2r\sin\theta+c_1r^2\cos^2\theta+2c_2r^2\cos\theta\sin\theta+c_3r^2\sin^2\theta\right.\\
&&\qquad\qquad+\frac{1}{2}\le.(b_1^2r^2\cos^2\theta+2b_1b_2r^2\cos\theta\sin\theta+b_2^2r^2\sin^2\theta)+O(r^3)\right\}.
\end{eqnarray*}
By a straightforward calculation, we have
\begin{eqnarray}\nonumber
\int_{B_\delta(p_i)\setminus B_{2R\epsilon}(p_i)}e^{\phi_\epsilon}dv_g&=&\int_{B_\delta(p)\setminus B_{2R\epsilon}(p)}
e^{\widetilde{G}_p}dv_g\\\nonumber
&=&\pi e^{\widetilde{A}_p}\le\{(2R\epsilon)^{-2}-(c_1+c_3+\frac{b_1^2+b_2^2}{2})\log(R\epsilon)\right.\\\label{integ-3}
&&\qquad\qquad+\le. \frac{K(p)}{3}\log(R\epsilon)+O(1)\right\}.
\end{eqnarray}
and
\begin{eqnarray}\nonumber
\int_{B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}e^{\phi_\epsilon}dv_g&=&
\int_{B_{2R\epsilon}(p)\setminus B_{R\epsilon}(p)}e^{\widetilde{G}_p-\eta_1\beta_1}dv_g\\
&=&\pi e^{\widetilde{A}_p}((R\epsilon)^{-2}-(2R\epsilon)^{-2}+O(1)).\label{integ-4}
\end{eqnarray}

For $y\in\mathbb{B}_{\delta}(0)$, we write
$$h(\exp_p(y))-h(p)=k_1r\cos\theta+k_2r\sin\theta+k_3r^2\cos^2\theta+2k_4r^2\cos\theta\sin\theta+k_5r^2\sin^2\theta+O(r^3).$$
Noting that $h(p_i)=h(p)$, $i=1,\cdots,\ell$, we have
\begin{align}\label{h-1}
&\int_{B_{R\epsilon}(p_i)}(h-h(p_i))e^{\phi_\epsilon}dv_g\nonumber\\
=&\int_{B_{R\epsilon}(p)}(h-h(p))e^{\phi_\epsilon}dv_g\nonumber\\
=&\le(\frac{1}{8}+\frac{1}{R^2}\right)^2e^{\widetilde{A}_p}\epsilon^{-2}\le\{32\pi (k_3+k_5+k_1b_1+k_2b_2)\epsilon^2\log\le(1+{R^2}/{8}\right)+O(\epsilon^2)\right\}.
\end{align}
Similarly
\begin{equation}\label{h-2}\int_{B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}(h-h(p_i))e^{\phi_\epsilon}dv_g=
\int_{B_{2R\epsilon}(p)\setminus B_{R\epsilon}(p)}(h-h(p))e^{\phi_\epsilon}dv_g=O(1)\end{equation}
and
\begin{eqnarray}\nonumber\int_{B_{\delta}(p_i)\setminus B_{2R\epsilon}(p_i)}(h-h(p_i))e^{\phi_\epsilon}dv_g&=&
\int_{B_{\delta}(p)\setminus B_{2R\epsilon}(p)}(h-h(p))e^{\phi_\epsilon}dv_g\\\label{h-3}
&=&-\pi e^{\widetilde{A}_p}(k_3+k_5+k_1b_1+k_2b_2)\log{(R\epsilon)}+O(1).\end{eqnarray}
It follows from (\ref{integ-1}) and (\ref{h-1}) that
\begin{eqnarray}\nonumber
\int_{B_{R\epsilon}(p_i)}he^{\phi_\epsilon}dv_g&=&h(p_i)\int_{B_{R\epsilon}(p_i)}e^{\phi_\epsilon}dv_g+
\int_{B_{R\epsilon}(p_i)}(h-h(p_i))e^{\phi_\epsilon}dv_g\\\nonumber
&=&\pi h(p)e^{\widetilde{A}_p}\epsilon^{-2}\le\{\frac{R^2+8}{8R^2}
+16\le(b_1^2+b_2^2-\frac{2}{3}K(p)+2\frac{k_3+k_5+k_1b_1+k_2b_2}{h(p)}\right)\right.\\\label{ann-0}
&&\le.\times\le(\frac{1}{8}+\frac{1}{R^2}\right)^2\epsilon^2\log\le(1+\frac{R^2}{8}\right)
+O(\epsilon^2)\right\}.
\end{eqnarray}
In view of (\ref{integ-4}) and (\ref{h-2}),
\begin{equation}\label{ann-1}\int_{B_{2R\epsilon}(p_i)\setminus B_{R\epsilon}(p_i)}he^{\phi_\epsilon}dv_g
=\pi h(p)e^{\widetilde{A}_p}((R\epsilon)^{-2}-(2R\epsilon)^{-2}+O(1)).\end{equation}
Also we have by (\ref{integ-3}) and (\ref{h-3}) that
\begin{eqnarray}\nonumber\int_{B_\delta(p_i)\setminus B_{2R\epsilon}(p_i)}he^{\phi_\epsilon}dv_g&=&
\pi h(p)e^{\widetilde{A}_p}\le\{(2R\epsilon)^{-2}-\le(c_1+c_3-\frac{K(p)}{3}+\frac{b_1^2+b_2^2}{2}\right.\right.\\
&&\quad\le.\le.+\frac{k_3+k_5+k_1b_1+k_2b_2}{h(p)}\right)\log(R\epsilon)+O(1)\right\}.\label{ann-2}\end{eqnarray}
Clearly
$$\int_{\Sigma\setminus\cup_{i=1}^\ell B_\delta(p_i)}he^{\phi_\epsilon}dv_g=
\int_{\Sigma\setminus\cup_{i=1}^\ell B_\delta(p_i)}he^{\widetilde{G}_p}dv_g=O(1),$$
which together with (\ref{ann-0})-(\ref{ann-2}) leads to
\begin{eqnarray}\nonumber
\int_\Sigma he^{\phi_\epsilon}dv_g&=&\frac{\pi\ell h(p)e^{\widetilde{A}_p}}{8\epsilon^2}\le\{
1+\frac{16}{R^2}+128\le(b_1^2+b_2^2-\frac{2}{3}K(p)+2\frac{k_3+k_5+k_1b_1+k_2b_2}{h(p)}\right)\right.\\\nonumber
&&\quad\quad\qquad\times\le(\frac{1}{8}+\frac{1}{R^2}\right)^2\epsilon^2\log\le(1+\frac{R^2}{8}\right)-8\le(
c_1+c_3-\frac{K(p)}{3}+\frac{b_1^2+b_2^2}{2}\right.\\\label{int-S}
&&\quad\quad\qquad \le.\le.+\frac{k_3+k_5+k_1b_1+k_2b_2}{h(p)}\right)\epsilon^2\log(R\epsilon)+O(\epsilon^2)\right\}.
\end{eqnarray}
Now we choose $R$ such that $R^4\epsilon^2=\frac{1}{\log(-\log{\epsilon})}$. It follows that $R^{-4}=o(\epsilon^2\log\epsilon)$
and $\log R/\log\epsilon=-1/2+o(1)$ as $\epsilon\rightarrow 0$. This together with (\ref{int-S}) implies that
\begin{eqnarray}\nonumber
\log\int_\Sigma he^{\phi_\epsilon}dv_g&=&\log\frac{\pi\ell h(p)e^{\widetilde{A}_p}}{8\epsilon^2}+\frac{16}{R^2}+4\le\{
b_1^2+b_2^2-\frac{2}{3}K(p)+c_1+c_3+\right.\\
&&\qquad\qquad \le.+2\frac{k_3+k_5+k_1b_1+k_2b_2}{h(p)}+o_\epsilon(1)\right\}\epsilon^2\log\frac{1}{\epsilon}.\label{log-1}
\end{eqnarray}
In view of (\ref{energy}),
\begin{eqnarray}\nonumber
\frac{1}{2}\int_\Sigma|\nabla_g\phi_\epsilon|^2dv_g&=& 16\pi\ell\log\frac{1}{\epsilon}-8\pi\ell\log 8-8\pi\ell+4\pi\ell \widetilde{A}_p+\frac{128\pi\ell}{R^2}\\
&&\qquad\quad+\le(\frac{64}{3}\pi\ell K(p)+o_\epsilon(1)\right)\epsilon^2\log\frac{1}{\epsilon}.
\label{energy-1}
\end{eqnarray}
Moreover, (\ref{average}) implies that
\begin{equation}\label{average-1}\overline{\phi}_\epsilon=\le(-\frac{16\pi\ell}{{\rm Vol}_g(\Sigma)}+o_\epsilon(1)\right)\epsilon^2\log\frac{1}{\epsilon}.\end{equation}
Noting that $2(k_3+k_5)=-\Delta_gh(p)$ and combining (\ref{log-1})-(\ref{average-1}) and Lemma \ref{c1-3-k}, we conclude
\begin{eqnarray}\nonumber
J_{8\pi\ell}(\phi_\epsilon-\overline{\phi}_\epsilon)
&=&\frac{1}{2}\int_\Sigma|\nabla_g\phi_\epsilon|^2dv_g-8\pi\ell\log
\int_\Sigma he^{\phi_\epsilon}dv_g+8\pi\ell \overline{\phi}_\epsilon\\\nonumber
&=&-8\pi\ell-4\pi\ell \widetilde{A}_p-8\pi\ell \log(\pi\ell h(p))-32\pi\ell\le\{\frac{8\pi\ell}{{\rm Vol}_g(\Sigma)}-2K(p)\right.\\
&&\quad\le. +b_1^2+b_2^2-\frac{\Delta_g h(p)}{h(p)}+\frac{2(k_1b_1+k_2b_2)}{h(p)}+o_\epsilon(1)\right\}\epsilon^2\log\frac{1}{\epsilon}.\label{hy-c}
\end{eqnarray}
Under the hypothesis (\ref{hy-2}), one can see for (\ref{hy-c}) that
$$J_{8\pi\ell}(\phi_\epsilon-\overline{\phi}_\epsilon)< -4\pi\ell
(2\log(\pi\ell h(p))+\widetilde{A}_p)-8\pi\ell.$$
This together with (\ref{ma-x}) implies (\ref{test}) and completes the proof of Theorem \ref{Thm3}. $\hfill\Box$

\section{Geometric conditions}\label{Sec6}

In this section, we give examples of surfaces on which the assumptions of Theorems \ref{Thm2} and \ref{Thm3} may be satisfied.
As a consequence, on such surfaces the mean field equation (\ref{meanfield}) may have a solution for appropriate
$h$.

\subsection{Geometric condition for Theorem \ref{Thm2}}
For $x=(x_1,x_2)$, $y=(y_1,y_2)\in \mathbb{R}^2$,
$x\sim y$ if and only if $x_1=y_1+m$ and $x_2=y_2+k$ for some integers $m$ and $k$. Define $\tau:\mathbb{R}^2\rightarrow \mathbb{T}^2=\mathbb{R}^2/\sim$
by $\tau(x)=[x]$, the equivalent class of $x$. If $U$ is an open subset of $\mathbb{R}^2$, then $\tau(U)$ is defined as an open subset
of the torus $\mathbb{T}^2$.
Let $g={(\tau^{-1})}^{\ast}(g_0)$ be a Riemannian metric on $\mathbb{T}^2$, where $g_0=d{x_1}^2+d{x_2}^2$ is the standard Euclidean metric. Then $(\mathbb{T}^2,g)$ is a flat torus.
Let $\tau_0: \mathbb{R}^2\rightarrow\mathbb{R}^2$ be a transition given by $\tau_0(x_1,x_2)=(x_1+1/2,x_2)$ for all $(x_1,x_2)\in\mathbb{R}^2$,
We set $\textbf{G}=\{\sigma_1,\sigma_2\}$, where $\sigma_1(P)=P$ and
$\sigma_2(P)=\tau\circ\tau_0\circ\tau^{-1}(P)$ for all $P\in\mathbb{T}^2$. Then $\textbf{G}$ is an isometric group acting on $(\mathbb{T}^2,g)$.
Clearly $I(P)=\sharp \textbf{G}(P)=2$ for all $P\in\mathbb{T}^2$. Motivated by Ding-Jost-Li-Wang \cite{DJLW}, we define a function on $\mathbb{R}^2$ by
\begin{eqnarray*}\lambda(x)&=&4\pi\le({x_2}^2-x_2+\frac{1}{6}\right)-2\log\le(1+e^{-4\pi x_2}-2e^{-2\pi x_2}\cos(2\pi x_1)\right)\\
&&-\sum_{n=1}^\infty\log
\le(1+e^{-4\pi n-4\pi  x_2}-2e^{-2\pi n-2\pi  x_2}\cos(2\pi x_1)\right)\\&&-\sum_{n=1}^\infty\log
\le(1+e^{-4\pi n+4\pi  x_2}-2e^{-2\pi n+2\pi  x_2}\cos(2\pi x_1)\right).\end{eqnarray*}
According to (\cite{Lang}, pages 42-47), we have $\lambda(x_1+1,x_2)=\lambda(x_1,x_2)$ and
$\lambda(x_1,x_2+1)=\lambda(x_1,x_2)$. Moreover, for any fixed $P\in\mathbb{T}^2$, if we assume $0\in\tau^{-1}(P)$, then
$G(P,Q)=\lambda\circ \tau^{-1}(Q)$ is the usual Green function on $\mathbb{T}^2$. Precisely $G$ satisfies $G(P,Q)=G(Q,P)$ for all
$P,Q\in \mathbb{T}^2$ and
$$\le\{\begin{array}{lll}\Delta_{g,Q}G(P,Q)=8\pi\delta_P(Q)-8\pi\\[1.5ex]
\int_{\mathbb{T}^2}G(P,Q)dv_{g,Q}=0.\end{array}\right.$$
It was proved by Ding-Jost-Li-Wang \cite{DJLW} that $G(P,Q)$ has the following decomposition near $P$
\begin{equation}\label{totic}G(P,Q)=-4\log r+A_P+O(r^2),\nonumber \end{equation}
where $r$ denotes the geodesic distance between $Q$ and $P$ and
\begin{equation}\label{AP}A_P=-4\log (2\pi)+\frac{2\pi}{3}-8\sum_{n=1}^\infty\log(1-e^{-2\pi n}).\nonumber\end{equation}
Let $\widetilde{G}_P(Q)=G(P,Q)+G(\sigma_2(P),Q)$ for all $Q\in\mathbb{T}^2$. Then $\widetilde{G}_P$ is a $\textbf{G}$-invariant Green function
satisfying
$$\le\{\begin{array}{lll}\Delta_{g}\widetilde{G}_P(Q)=8\pi\le(\delta_P(Q)+\delta_{\sigma_2(P)}(Q)\right)-16\pi\\[1.5ex]
\int_{\mathbb{T}^2}\widetilde{G}_P(Q)dv_{g}=0.\end{array}\right.$$
By elliptic estimates, we can write near $P$
$$\widetilde{G}_P(Q)=-4\log r+\widetilde{A}_P+O(r).$$
Clearly
\begin{equation}\label{tildeAP}\widetilde{A}_P=A_P+G(\sigma_2(P),P)=A_P+G(P,\sigma_2(P)).\end{equation}

Using the inequality $\log(1+t)\geq t/(t+1)$ for $t>-1$, we have
\begin{eqnarray}\nonumber
A_P+2+2\log\pi&\leq&2-4\log 2-2\log\pi+\frac{2\pi}{3}+8\sum_{n=1}^\infty\frac{e^{-2\pi n}}{1-e^{-2\pi n}}\\\nonumber
&\leq& 2-4\log 2-2\log\pi+\frac{2\pi}{3}+\frac{8}{1-e^{-2\pi}}\sum_{n=1}^\infty {e^{-2\pi n}}\\\nonumber
&=&2-4\log 2-2\log\pi+\frac{2\pi}{3}+\frac{8e^{-2\pi}}{(1-e^{-2\pi})^2}\\\label{approx-1}
&\approx& -0.9752.
\end{eqnarray}
Similarly
\begin{eqnarray}\nonumber
G(P,\sigma_2(P))+2\log 2&=&\lambda(\frac{1}{2},0)+2\log 2\\\nonumber
&=&\frac{2\pi}{3}-2\log 2-8\sum_{n=1}^\infty\log(1+e^{-2\pi n})\\\nonumber
&\leq&\frac{2\pi}{3}-2\log 2-8\sum_{n=1}^\infty \frac{e^{-2\pi n}}{1+e^{-2\pi n}}\\\nonumber
&\leq&\frac{2\pi}{3}-2\log 2-\frac{8e^{-2\pi}}{1-e^{-4\pi}}\\\label{approx-2}
&\approx&0.6932.
\end{eqnarray}
In view of (\ref{tildeAP}), we have by combining (\ref{approx-1}) and (\ref{approx-2}) that
$\widetilde{A}_P<-2-2\log\pi-2\log 2$.
Since $\widetilde{A}_P$ is independent of the base point $P$,  we conclude
\begin{equation}\label{maxim}\max_{P\in\mathbb{T}^2}\widetilde{A}_P<-2-2\log\pi-2\log 2.\end{equation}
Since ${\rm Vol}_g(\mathbb{T}^2)=1$, (\ref{maxim}) is exactly (\ref{cond}) where $h$ is a constant and $\ell=2$.

In conclusion, we construct a flat torus $(\mathbb{T}^2,g)$ with the action of an isometric group $\textbf{G}=\{\sigma_1,\sigma_2\}$
such that (\ref{cond}) holds when $h$ is a positive constant. Set $h=c+\epsilon\phi$, where $c>0$ and $\epsilon>0$ are two constants,
$\phi:\mathbb{T}^2\rightarrow \mathbb{R}$ is a smooth function satisfying $\phi(\sigma_1(x))=\phi(\sigma_2(x))$ for all $x\in\mathbb{T}^2$.
If $\epsilon$ is chosen sufficiently small, then (\ref{cond}) still holds.

\subsection{Geometric condition for Theorem \ref{Thm3}}

Let $\{\mathbf{a},\mathbf{b}\}$ be a basis on $\mathbb{R}^2$.
Consider a torus $\mathbb{T}^2=\mathbb{R}^2/\sim$, where $x\sim y$ if and only if $x-y=k\mathbf{a}$ for some integer $k$ or
$x-y=m\mathbf{b}$ for some integer $m$. Let $g$ be a flat metric induced by the standard Euclidean metric on $\mathbb{R}^2$.
Denote the Gaussian curvature of $(\mathbb{T}^2,g)$ by $K$. Then $K(x)\equiv 0$ for all $x\in\mathbb{T}^2$. For any fixed positive
integer $\ell$, we set $\sigma_1(x)=x$, $\sigma_2(x)=x+\frac{1}{\ell}\mathbf{a}$, $\cdots$, $\sigma_\ell(x)=x+\frac{\ell-1}{\ell}\mathbf{a}$
for all $x\in \mathbb{T}^2$. Obviously $\textbf{G}=\{\sigma_1,\cdots,\sigma_\ell\}$ is an isometric group acting on $(\mathbb{T}^2,g)$.
One can easily see that (\ref{hy-2}) holds when $h$ is a positive constant. As in the previous subsection, we set
$h=c+\epsilon\phi$, where $c>0$ and $\epsilon>0$ are two constants,
$\phi:\mathbb{T}^2\rightarrow \mathbb{R}$ is a smooth function satisfying $\phi(\sigma(x))=\phi(x)$ for all $\sigma\in\textbf{G}$ and
all $x\in\mathbb{T}^2$.
If $\epsilon$ is chosen sufficiently small, then (\ref{hy-2}) still holds.\\

{\bf Acknowledgements}. This work is partly supported by the National Science Foundation of China (Grant Nos.
  11471014 and  11761131002).

redpajama_set_name: RedPajamaArXiv


text:
African Visual Artists Raising Mental Health Awareness
"The Value of Nothing", 2018 | Courtesy of the artist
Lize Okoh
To mark World Mental Health Day, Culture Trip celebrates the work of five visual artists from Africa whose work creates awareness surrounding mental health issues on the continent.
Mental health problems often go untreated, with as many as one in four of us suffering from them each year. Often it's difficult to tell when our mental health is in jeopardy. In most cases it's the stigma that prevents people from speaking out or seeking support. This discrimination towards mental health issues can be found across Africa, where it's often the older generation who see mental illnesses as a 'Western disease'.
But times are changing, and a new generation seeks different ways to better understand the human experience. Artists of varied disciplines are using their skills and creativity to address mental health issues and to help others heal through the power of art and expression. Here are five visual artists from Africa who are taking on the challenge of raising awareness of mental health issues.
Nosipho Nxele
Nosipho Nxele is a self-taught illustrator and art director based in Johannesburg, South Africa. In her 2016 series Mental Squad, she discusses depression, anxiety and spirituality and uses a surrealist illustration technique to demonstrate the different stages of depression a woman can go through. It was based on her own experience of long unresolved mental health issues that left her bitter, broken and panic-stricken. Her use of colours was specific to the piece: the pink background represents a woman's innocence, the black the cloud of depression and the white the peace the victim longs for. Nxele now sees the creative process of illustrating as a means of healing and repairing the deep emotional scars left by her depression.
Rip me open because I swear there is some good in me. I refuse to be a reject…second best and never the chosen one. I know there is light at the end of the tunnel. #illustration #surealism #depression #mentalillness
A post shared by Nosy (@just.illustrat) on Aug 10, 2017 at 5:39am PDT
Eloghosa Osunde
Eloghosa Osunde is a Nigerian writer and visual artist whose work revolves around the topics of mental health, sexuality and identity. She's an alumnus of the Farafina Creative Writing Workshop (2015) and the Caine Prize Workshop (2018), and her visual series And Now We Have Entered Broken Earth has been exhibited across the globe and was selected for the New York Times Portfolio Review and Photoville's EmergiCubes. In Color This Brain, a six-part photographic series that demonstrates her consciousness of mental health, she sees her different emotional states in various colours and tones. She says of the project, "I know the exact colour and texture of my depression and my anxiety, my dissociation and my exhaustion. I know the flatness and madness and sharpness of all, the pace at which they breathe behind the eyes."
Nobukho Nqaba
Nobukho Nqaba is a 26-year-old performance artist from Butterworth in the Eastern Cape in South Africa. She graduated from the Michaelis School of Fine Art in Cape Town in 2012, and her work is inspired by deeply personal experiences. In 2016 Nqaba performed Ndiyayekelela (Letting Go) at the Also Known as Africa art fair in Paris. Although not directly addressing the single issue of mental health, her performance relays her feelings of loss and identity as she struggles to come to terms with the loss of her father. Nqaba uses objects such as overalls and blankets to symbolise the belongings her father left behind and to address an internal struggle as she wrestles with feelings of guilt, fondness and confusion.
'Ndinikezela (Battle)' performed by Nobukho Nqaba | Courtesy of the artist
Ken Nwadiogbu
Ken Nwadiogbu is a 24-year-old multidisciplinary artist from Nigeria whose hyperrealistic drawings illustrate global events and their effects on the human experience. The self-taught artist sees his work as being an emotive means of creating discussion surrounding sociopolitical structures and issues in society. His artistry is driven by a desire to change the way people think about themselves and the society in which they live. One of his pieces, inspired by the #MeToo movement, was even covered by CNN Africa. After receiving so much positive feedback on his incredibly poignant composition, he realised his role as "an activist on canvas", lending his creative hand to uplift the voices of those who are shunned or overlooked by society.
"Me Too" 2018 | Courtesy of the artist
With his realistic drawing The Value of Nothing, Nwadiogbu challenges his audience to see depression and suicidal thoughts as what happens when we base our value on what other people think of us, and a lack of self-awareness and self-appreciation.
Courage in Congo Project
The Courage in Congo project was started in 2015 as a 14-week psychosocial programme for a community of adolescent girls living in Goma, in the eastern Democratic Republic of the Congo. It was directed by an arts educator and a specialist in behavioural health sciences who sought to help improve the mental health of these girls, who live in a community affected by prevalent sexual and gender-based violence. Together Christina Mallie and Nadia Fazal studied the community leaders' perceptions of girls and their role in society and worked towards changing these perceptions through art and positive affirmation. With the belief that art has a therapeutic and transformative power, the girls decorated the walls of their neighbourhoods with murals based on the themes 'Women in the Workforce' and 'The Development and Promotion of Women in Leadership'.
A section of the "Development and Promotion of Women Leadership" mural, April 2016 | Courtesy of Colors of Connection
The participants and project staff in front of the "Women in the workforce" mural, April 2016. | Courtesy of Colors of Connection
Wednesday, 10 October is World Mental Health Day. To highlight this, Culture Trip is looking at how different societies are shining a light on this important issue in innovative and alternative ways.
The content of this article is provided for general information only and is not an attempt to practise medicine or give specific medical advice, including, without limitation, advice concerning the topic of mental health. The information contained in this article is for the sole purpose of being informative and is not to be considered complete, and does not cover all issues related to mental health. Moreover, this information should not replace consultation with your doctor or other qualified mental health providers and/or specialists. If you believe you or another individual is suffering a mental health crisis or other medical emergency, please seek medical attention immediately.
If you are experiencing mental health issues in the UK, Samaritans can be contacted on 116 123 or jo@samaritans.org. You can contact the mental health charity Mind by calling 0300 123 3393 or visiting mind.org.uk. Please note there are no affiliations of any kind between the aforementioned organisations and Culture Trip.

redpajama_set_name: RedPajamaCommonCrawl


I started making sounds for games just as Game Audio was about to enter the digital age. After creating sound drivers for Sound Blaster (FM) cards on a few games I became the digital audio expert, using my sampling skills to create 8 bit digital sounds for CPU Bach, Colonization and Civilization for Windows and Mac. Digidesign's Sound Designer was turning into Pro Tools and had a whopping 4 channels of playback. Digital meant recording and editing VO and creating Soundtracks for Movies and Logos as well.
Four channels turned to eight, and then Sixteen. Dolby started getting interested in games, eager to see the quality improve. Movies were now mixed in Pro Logic - a matrix surround with a center channel and 2 rear speakers which eventually became 2 independent rear speakers, paving the way for 5.1 Mixes. Matrix encoding brought steering issues into the mix, making monitoring a ProLogic II mix in real time mandatory.
Audio that is synthesized, never sounds organic. We're getting closer and we might get there some day. I hope to live to hear it.


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Civil Courts in Dumont
Courts in Dumont, New Jersey
Find the right Litigation attorney in Dumont, NJ
Litigation Lawyers in Dumont
The courts of Dumont, New Jersey were formed by the government of New Jersey to assist residents of Dumont resolve legal disagreements which they cannot settle amongst themselves.
More likely than not, you will interact with the courts of Dumont, New Jersey at least once or twice in your life.
Dumont, New Jersey's courts oversee both civil and criminal matters. The litigation lawyers of Dumont, New Jersey spend a good deal of time in the courts, and normally know the ins and outs of the local court system pretty well. However, to a layperson, dealing with the court system for the first time can be intimidating. Here are a few of the most likely situations in which a person will have to deal with the courts in Dumont, New Jersey:
Events Leading To Exposure To The Courts In Dumont, New Jersey
Jury Duty: If you're a citizen of the United States, and live in Dumont, you've probably already interacted with the court system of Dumont, New Jersey by being called to jury duty. The law requires you to show up for jury duty if you are called to do so. This involves receiving a letter informing you that you have jury duty, and telling you when and where you need to show up. On the appointed day, you will be placed in a "juror pool," where you will wait to be called into court for an upcoming trial. The lawyers for both sides of the case will then engage in jury selection. If you are eliminated from the juror pool, your service is complete. If you are selected to serve on a jury, you will have to show up for the entire trial, or you might face criminal charges.
Filing a lawsuit: When you sue a person or company in a Dumont, New Jersey court, it's a given that you will have to interact with the courts, to one degree or another. At the very least, you'll probably have to attend some hearings, deal with discovery disagreements, and attend case management meetings. Even if your case never goes to trial (which it probably won't, statistically), you will probably become quite familiar with Dumont, New Jersey's courts.
Being Sued: If you face the unenviable prospect of getting sued in Dumont, New Jersey, you'll be spending a good deal of time dealing with the courts. You will have to file an answer to the complaint, discovery requests, motions, and many other documents with the court. Majority of these will happen whether or not the case even goes to trial.
Divorce: Most divorces in Dumont, New Jersey are fairly quick, and don't require a lot of involvement from the courts. Some, however, can get combative, and the local court system may be obligated to resolve disputes between the parties.
How Can A Dumont, New Jersey Tort Lawyer Help?
Most people don't have to go to court in Dumont, New Jersey unless they're dealing with some confusing legal issues.
You will probably need assistance navigating these legal issues. You should definitely consider hiring the services of a knowledgeable litigation lawyer in Dumont, New Jersey if you believe that you are going to have significant contact with the court system sometime soon.
Dumont Civil Depositions Attorneys
Dumont Commercial Litigation Lawyers
Dumont Punitive Damages Attorneys
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Dumont Tort Lawyers Lawyer
Litigation Lawyers in Cinnaminson
Litigation Lawyers in Bellmawr
Litigation Lawyers in Englewood
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Litigation Lawyers in Plainfield
Litigation Lawyers in Perth Amboy


Each month, Seattle Sperm Bank selects an outstanding donor from our catalog to be our Featured Donor. Along with highlighting his physical attributes and character traits, we also offer a special 2 vials for the price of 1 deal for Featured Donors.
We choose featured donors based on two factors. The first, which is objective, is whether the donor has enough vials available to qualify for our 2 for 1 offer. The second factor is subjective — we select a donor we think is better than average and worthy of bringing to the attention of our clients. It's just one of the benefits SSB provides. We're small enough to get to know all our donors while also providing you with individual care and attention so you can find a donor that's right for you.
Our current Featured Donor is Smith, whose specimens came out of quarantine and became available in September. Along with having enough supply to qualify for our special offer, Smith is an exceptional donor! Smith is an analytical thinker, smart, hard-working, and athletic. He is incredibly loyal to his family and close friends, and loves adventuring outdoors. Smith is also quite handsome, with thick wavy blonde hair and stunningly bright blue eyes. We think he deserves more attention from our clients, and we encourage you to view his profile. But don't delay! Our Featured Donors usually sell out fast, so if Smith is the donor of your choice, we recommend you make your decision soon.
If you have questions about a Featured Donor, or any other part of the process, please contact one of our Client Services coordinators at 206-588-1484 or cs@seattlespermbank.com.


text:
Staycation UK: The best places to stay this summer
Author: Joe White | Updated: 12 May 2021 12:19
Having free reign of international travel is not looking likely this year, so where is the best place for a UK staycation? Well, we've got the lowdown from Ferrari to help answer that question - from the UK's own version of Route 66 to where Netflix's award-winning The Crown was filmed.
We understand that not everybody will be able to cruise around in a 3.9-litre V8 turbo Ferrari Portofino M, but that shouldn't stop you exploring the UK in whatever car you drive.
Honister Pass, Lake District
The Lake District's finest uninterrupted driving is to be found on the Honister Pass, which is the section of the B5289 between Seatoller and Gatesgarth. The Honister Pass is only 3.7 miles long, but if you start in Keswick and take the B5289 south past Derwent Water, follow the Honister Pass west and take the right hand turn north up the Newlands Pass to Braithwaite, then take the A66 east for a few minutes back to Keswick, you'll have racked up 23 miles of smiles.
The Pass skirts the Honister Slate Mine, the last working mine of its kind in England. It's also a tourist attraction, offering underground tours, climbing and abseiling.
Something the Lake District is not short of is fine dining. There are more Michelin-starred restaurants here per capita than London, Paris and Tokyo combined, as evidenced in Steve Coogan and Rob Brydon's comi-travel series The Trip. Top of the list is L'Enclume.
You may also wish to pay your respects to the region's most famous speed demon. Donald Campbell CBE set the world water speed record on Ullswater in 1955 at an average speed of 202.32mph. He would die in 1967 at Coniston Water, aged 45, trying to go faster still, and is now buried in the churchyard. Campbell's headstone is easy to spot - there's a bluebird on it.
A686, Cumbria and Northumberland
The Pennines never disappoint, and the A686 is a particularly majestic marriage of geography and engineering, full of zig-zagging and gear-changing joy throughout. You will insist on manual-mode here.
The beguiling 37 miles that connect Penrith, Cumbria, with the village of Haydon Bridge in Northumberland ranks as one of the north of England's top three bits of blacktop. Crossing the River Eden, the road starts climbing and the landscape begins to brown from Melmerby to 580m above sea level at the Hartside summit, providing views across the Solway Firth and to Scotland.
The well-sighted road leads to Alston, a charming village with quaint cobbled streets which claims to be England's highest market town (they're locked in a bragging battle with Buxton). The A686 beyond Alston cruises through another astonishing panorama of windswept upland before blasting down to the River Allen's wooded gorge and throwing in some more stellar switchbacks for good measure.
Northumberland's most acclaimed hotel of the moment is the Lord Crewe Arms, 20 miles south east of Haydon Bridge, in the honey-hued village of Blanchland. Its beauty has drawn poets including WH Auden and Philip Larkin. The fabric of the building dates back to a 12th century priory. Blanchland Abbey was lauded for its hospitality for 400 years until its dissolution in 1539. It passed through different private ownerships and from the 1720s, as the Lord Crewe Arms, served pints to miners after their shift.
The pub part is now housed in the vaulted crypt. Food is worshipped upstairs, either in the Bishop's dining room or beside a fireplace big enough to toast the toes of his entire flock. It's so large that Tory politician, landowner and rebel 'General' Tom Foster hid inside it during the 1715 Jacobite rising, and the hotel is said to be haunted by his sister, Dorothy.
Black Mountain Road, Brecon Beacons
So wild is the Brecon Beacons' wilderness, that this is where the SAS train. The roads which criss-cross it are as majestic as the land itself. There's the B4560, A470, A470, A4059, A4067, and best of all the A4069, aka Black Mountain Road.
Located in the western area of the national park, it's famous for its twists, dips and climbs, and is known to many petrolheads as 'the Top Gear road'. Stretching from Upper Brynamman up to Llangadog, it's 20 miles long and reaches a height of 493m above sea level. From a driving perspective, the road is best tackled from north to south, especially the hairpin known as Tro Gwcw, or 'Cuckoo Turn', but the breathtaking views of the Tywi Valley are best appreciated going the other way, so you should turn around and come back the way you came.
Located on the A465 in the small village of Llanfihangel Crucorney, the 12th century Skirrid Mountain Inn is said to be the most haunted pub in the country, having once been a rallying point for those revolting against the rule of Henry IV.
North Coast 500
There are many roads around the world capable of vying for the title World's Greatest. From the scenically spectacular - Australia's Great Ocean Road, say, or California's Pacific Coast Highway; and the exciting - Italy's Stelvio Pass and Mount Fuji's Touge roads; through to the dangerous - Bolivia's Yungas Road or, for that matter, Germany's killer Nürburgring. But we'll wager that the most joyfully sublime route of them all is right here in the UK: Scotland's northernmost coastal and highland roads.
It's become known as the North Coast 500. Some call it Scotland's Route 66; a 516-mile loop from Inverness to the tip of the isle and circumnavigating back. It's popular in Summer with supercars, caravans, bikers and cyclists, each tackling it at their own pace but pausing regularly to absorb the cracking views.
Drive the route anticlockwise and you'll find the scenery just gets more and more spectacular. Take the A9 out of Inverness, cross the Black Isle, named after its dark farming soil, and pass the Dalmore whisky distillery to make your way up the east coast.
Pass the turreted Dunrobin Castle and continue along the A9 until Dunbeath. Take the A99 and keep an eye out for dolphins. At Wick, you may wish to visit the five-storey 15th century Ackergill Tower. Transporting you back in time, this castle (which has lately been used as a filming location for The Crown) is stocked with interesting antiques and furniture, taxidermy and tartan, and boasts an enchanting vista of coal-black rock stacks and crashing waves.
The scenery, unbelievably, continues to improve. Through South West Sutherland, you will pass the dramatic peaks of Canisp, Cul Mor and Ben More Assynt. This Geopark consists of some of the most ancient and unique geology on the planet, dating back three billion years. Park at Gruinard Bay, and admire its rocky coves and unusual pink sand leading out to island-studded waters.
The twistiest part of the journey is the Bealach na Ba mountain pass. The views across to the Isle of Skye here are incredible and it feels like you're on the edge of the world. At Applecross, look out for bobbing seals.
Careful of the fog that often descends on Beinn Bhan mountain, for the drops are sheer. Once back on the A832 you'll be able to put the hammer down and the threescore of these final 516-miles will flicker in the rear-view mirror.


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