Exact counting of spherical metrics with one conical singularity on rectangular tori
Abstract
We prove that for every integer n≥ 2 and 8π(n−1)<ρ<8πn, the singular Liouville equation Δu+eu=ρδ0 on a rectangular torus Eib=ℂ/(ℤ+i bℤ) has exactly n solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item Ei b admits no spherical metrics with a conical singularity of angle 2πϑ as long as ϑ is a positive odd integer. \item For every integer n≥ 1, Ei b admits exactly n spherical metrics with a conical singularity of angle 2πϑ for each ϑ∈ (2n−1, 2n+1). \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for ρ≤ 8π. We develop a unified approach for all ρ∈ (0,+∞)∖ 8πℕ≥ 1 by exploring the deep connection with the monodromy of the classical Lamé equation.
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BibTeX
@misc{chen2026exact,
title = {{Exact counting of spherical metrics with one conical singularity on rectangular tori}},
author = {Zhijie Chen and Shihong Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30174},
eprint = {2608.30174},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30174},
}
RIS
TY - UNPB
TI - Exact counting of spherical metrics with one conical singularity on rectangular tori
AU - Zhijie Chen
AU - Shihong Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30174
UR - https://arxiv.org/abs/2608.30174
AB - We prove that for every integer n≥ 2 and 8π(n−1)<ρ<8πn, the singular Liouville equation Δu+e^u=ρδ_0 on a rectangular torus E_ib=ℂ/(ℤ+i bℤ) has exactly n solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item E_i b admits no spherical metrics with a conical singularity of angle 2πϑ as long as ϑ is a positive odd integer. \item For every integer n≥ 1, E_i b admits exactly n spherical metrics with a conical singularity of angle 2πϑ for each ϑ∈ (2n−1, 2n+1). \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for ρ≤ 8π. We develop a unified approach for all ρ∈ (0,+∞)∖ 8πℕ_≥ 1 by exploring the deep connection with the monodromy of the classical Lamé equation.
ER -