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

Abstract

We study families of spherical metrics on the flat torus $E_{\tau}$ $=$ $\mathbb{C}/\Lambda_{\tau}$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6\pi$, and at $\pm p$ is $4\pi$. 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 $G(z;\tau)$ 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 $a=r+s\tau.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6\pi$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.