## Elliptic functions, Green functions and the mean field equations on tori.(English)Zbl 1207.35011

The Green functions on flat tori are shown to have either three or five critical points (Theorem 1.3). In this paper, proof is done via the study of the mean field equation
$\Delta u+ \rho e^u= \rho\delta_0,\quad \rho\in \mathbb{R}_+.$
This equation has solutions if $$\rho\neq 8m\pi$$, $$m\in\mathbb{N}$$ [C.-C. Chen and C.-S. Lin, Commun. Pure Appl. Math. 56, No. 12, 1667–1727 (2003; Zbl 1032.58010)]. If $$\rho= 8\pi$$, the mean field equation has solutions if and only if the Green function has critical points other than three (Th.1.1) and has at most one solution up to scaling (Theorem 1.2). Theorem 1.3 follows from these two theorems.
The authors say computer simulation suggests the following picture: Let $$\Omega_3$$ (resp. $$\Omega_5$$) be the subset of the moduli space $${\mathcal M}_1\cup\{\infty\}\cong S^2$$, $${\mathcal M}_1= {\mathcal H}/\text{SL}(2, \mathbb{Z})$$ which corresponds to tori with three (resp. five) critical points. Then $$\Omega_3\cup \{\infty\}$$ is a closed subset containing $$i$$, $$\Omega_5$$ is an open subset containing $$e^{\pi i/3}$$, both of them are simply connected and their common boundary $$C$$ is a curve homeomorphic to $$S^1$$ containing $$\infty$$. The extra critical points are split out from some half period point when the tori moves from $$\Omega_3$$ to $$\Omega_5$$ across $$C$$. Partial results in this direction are also obtained (Theorem 1.6).
Let $$T= \mathbb{C}/\{\mathbb{Z}\omega_1+ \mathbb{Z}\omega_2\}$$ be a flat torus, $$G(z,w)$$ is the Green function;
$-\Delta_z(G(z, w)= \delta_w(z)-{1\over |T|},\quad\int_T G(z,w)\,dA= 0.$
Since $$G(z,w)= G(z-w, 0)$$, $$G(z):= G(z,0)$$ is also called the Green function. There exists a constant $$C(\tau)$$, $$\tau= \omega_2/\omega_2$$, such that
$8\pi G(z)= {2\over |T|} \int_T \log|\wp(\xi)- \wp(z)|\,dA+ C(\tau).$
Then analysing the period integral
$F(z)= \int_L {\wp'(z)\over \wp(\xi)- \wp(z)}\,d\xi,$
where $$L$$ is a line segment in $$T$$ which is parallel to the $$\omega_1$$-axis, it is shown $$z= t\omega_1+ s\omega_2$$ is a critical point of $$G$$ if and only if
$\zeta(t\omega_1+ s\omega_2)= t\eta_1+ s\eta_2,\;\zeta(z)=- \int^z \wp,\quad \eta_1= \zeta(z+ \omega_i)- \zeta(z).$
Since $$G$$ is even, half periods $${\omega_1\over 2}$$, $${\omega_2\over 2}$$ and $${\omega_3\over 2}$$, $$\omega_3= \omega_1+ \omega_2$$ are critical points of $$G$$. It is shown for rectangular tori that $$G(z)$$ has only three critical points. But if $$\omega_1= 1$$, $$\omega_2= e^{\pi i/3}$$, then $${\omega_3\over 3}$$ also is a critical point.
After these preparations, the mean field equation is investigated in §3. If $$\rho= 4\pi\ell$$, $$\ell\in\mathbb{N}$$, any solution $$u$$ of the mean field equation takes the form
$u= c_1+ \log{|f'|^2\over (1+|f|^2)^2},$
on the whole $$\mathbb{C}$$, with a meromorphic function $$f$$ [C. C. Chen, C. S. Lin and G. Wang, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3, No. 2, 367–397 (2004; Zbl 1170.35413)]. $$f$$ is called a developing map of $$u$$. Since $$u_{zz}-{1\over 2} u^2_z$$ is the Schwarz derivative of $$f$$, two developing maps $$f$$ and $$\overline f$$ are related by
$\overline f= Sf= {pf-\overline q\over qf+\overline p},\quad S\in\text{PSU}(1).$
Possible transformation by an element of $$\text{PSU}(1)$$ is shown essentially to be
$f(z+ \omega_1)= e^{2i\theta_1} f(z),\quad f(z+ \omega_2)= e^{2i\theta+2} f(z).$
On the other hand,
$f(z)= \exp\Biggl(\int^z_0 {\wp'(z_0)\over \wp(\xi)- \wp(z_0)}\,d\xi\Biggr)$
is well-defined and satisfies $$f(z+\omega_j)= e^{2i\theta_j}f(z)$$, hence Theorem 1.1 is obtained. Unique existence of solutions of the mean field equation in the case $$\rho= 4\pi$$ is also proved by the same way (Theorem 3.2).
In §4, extending the isoperimetric inequality of Bol for domains in $$\mathbb{R}^2$$ with metric $$e^w|dx|^2$$ to the case when the metric becomes singular, the linearized equation
$\Delta\varphi+\rho e^u\varphi= 0,\quad \varphi(z)= \varphi(-z)$
of the mean field equation is shown to have only the trivial solution if $$\rho\in [4\pi, 8\pi]$$ (Theorem 4.1). By the continuation from $$\rho= 4\pi$$ to $$8\pi$$, Theorem 1.2 follows from Theorem 3.2 and Theorem 4.1. Hence Theorem 1.3 is proved.
Let $$\omega_1=1$$ and $$\omega_2= \tau={1\over 2}+ bi$$, $$b> 0$$. Then Theorem 1.6 asserts the followings:
1.
There exist $$b_0< {1\over 2}< b_1<{\sqrt{3}\over 2}$$, such that $${1\over 2}$$ is a degenerate critical point of $$G(z;\tau)$$ if and only if $$b= b_0$$ or $$b= b_1$$. $${1\over 2}$$ is a local minimum point of $$G$$ if $$b_0< b< b_1$$ and it is a saddle point if $$b< b_0$$ or $$b> b_1$$.
2.
Both $${\tau\over 2}$$ and $${1+ \tau\over 2}$$ are nondegenerate saddle points of $$G$$.
3.
$$G$$ has two critical points $$\pm z_0(\tau)$$ when $$b< b_0$$ or $$b> b_1$$. They are nondegeneerate global minimum points of $$G$$ and in the former case, $$\text{Re\,}z_0(\tau)= {1\over 2}$$; $$0<\text{Im\,}z_0(\tau)< {b\over 2}$$.
To prove Th.1.6, the following theta function description of $$G(z)$$ is used
$G(z)=-{1\over 2\pi}\log|\vartheta_1(z)|+ {1\over 2b} y^2_ C(\tau).$
The function
$\vartheta_1(z; \tau)= -i\sum^\infty_{n=-\infty}(-1)^n q^{(n+1/2)^2} e^{(2n+1)\pi iz},$
$$q= e^\pi i\tau$$ satisfies the heat equation $${\partial^2\vartheta_1\over \partial z^2}= 4\pi i{\partial\vartheta_1\over\partial\tau}$$. By this fact, it is shown $$(\log\vartheta_1)_b$$ is decreasing from $$\infty$$ to $$-\pi/4$$ on the line $$L:{1\over 2}+ ib$$, $$b\in\mathbb{R}$$. Hence $$G_{xx}= 0$$ and $$G_{yy}= 0$$ occur exactly once on $$L$$, respectively (Theorem 8.1). Similar results for $$\vartheta_3$$ are also obtained (Theorem 9.1). By using these results and
$G\biggl({\omega_3\over 2}\biggr)- G\biggl({\omega_2\over 2}\biggr)= {1\over 4\pi}\log|\lambda(\tau)-1|,\quad \lambda(\tau)= {e_4- e_2\over e_1- e_2},$
(§5), Theorem 1.6 is proved in §6.

### MSC:

 35A08 Fundamental solutions to PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 30F99 Riemann surfaces 33E05 Elliptic functions and integrals 58J05 Elliptic equations on manifolds, general theory 58K05 Critical points of functions and mappings on manifolds

### Citations:

Zbl 1032.58010; Zbl 1170.35413
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### References:

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