Even Cone Spherical Metrics: Blow-Up at a Cone Singularity

Abstract

We study families of spherical metrics on the flat torus $E_τ$ $=$ $\mathbb{C}/Λ_τ$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6π$, and at $\pm p$ is $4π$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;τ)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+sτ.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6π$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.