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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2508.17604 |
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| _version_ | 1866917025943650304 |
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| author | Chen, Zhijie Fu, Erjuan Lin, Chang-Shou |
| author_facet | Chen, Zhijie Fu, Erjuan Lin, Chang-Shou |
| contents | Let $G(z)=G(z;τ)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics.
In this paper, firstly, we prove that once $G(z)$ has $5$ critical points, then these $5$ critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We prove that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of $G_p(z)$ as $p$ varies. Applications to the curvature equation $Δu+e^{u}=4π(δ_{p}+δ_{-p})$ on $E_τ$ are also given, and how the geometry of the torus affects the solution structure is studied. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Green functions, Hitchin's formula and curvature equations on tori Chen, Zhijie Fu, Erjuan Lin, Chang-Shou Analysis of PDEs Let $G(z)=G(z;τ)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics. In this paper, firstly, we prove that once $G(z)$ has $5$ critical points, then these $5$ critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We prove that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of $G_p(z)$ as $p$ varies. Applications to the curvature equation $Δu+e^{u}=4π(δ_{p}+δ_{-p})$ on $E_τ$ are also given, and how the geometry of the torus affects the solution structure is studied. |
| title | Green functions, Hitchin's formula and curvature equations on tori |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.17604 |