Exact counting of spherical metrics with one conical singularity on rectangular tori

Abstract

We prove that for every integer $n\geq 2$ and $8π(n-1)<ρ<8πn$, the singular Liouville equation $Δu+\e^u=ρδ_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2π\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2π\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \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&#39;s isoperimetric inequality only works for $ρ\leq 8π$. We develop a unified approach for all $ρ\in (0,+\infty)\setminus 8π\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lamé equation.