Shapes of tetrahedra with prescribed cone angles

Abstract:

Given real numbers $4\pi >\theta _0\geq \theta _1\geq \theta _2\geq \theta _3>0$ so that $\sum _{j=0}^3\theta _j=4\pi$, we provide a detailed description of the space of flat metrics on the 2-sphere with 4 conical points of cone angles $\theta _0,\theta _1,\theta _2,\theta _3$, endowed with a geometric structure arising from the area function.