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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2503.06935 |
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| _version_ | 1866910866850447360 |
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| author | Chen, Zhijie Fu, Erjuan Lin, Chang-Shou |
| author_facet | Chen, Zhijie Fu, Erjuan Lin, Chang-Shou |
| contents | Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ be a flat torus and $G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};τ):=\sum_{i<j}G(z_{i}-z_{j};τ)-n\sum_{i=1}% ^{n}G(z_{i};τ). \] Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb H=\{τ: \operatorname{Im}τ>0\}$ such that $G_n(\cdot;τ)$ has degenerate critical points for any $τ$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $τ\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;τ)$ are non-degenerate.
Applications to counting the exact number of solutions of the curvature equation $Δu+e^{u}=ρδ_{0}$ on $E_τ$
will be given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations Chen, Zhijie Fu, Erjuan Lin, Chang-Shou Analysis of PDEs Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ be a flat torus and $G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};τ):=\sum_{i<j}G(z_{i}-z_{j};τ)-n\sum_{i=1}% ^{n}G(z_{i};τ). \] Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb H=\{τ: \operatorname{Im}τ>0\}$ such that $G_n(\cdot;τ)$ has degenerate critical points for any $τ$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $τ\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;τ)$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $Δu+e^{u}=ρδ_{0}$ on $E_τ$ will be given. |
| title | Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.06935 |