On metrics of curvature 1 with four conic singularities on tori and on the sphere

TL;DR

This paper investigates conformal metrics of curvature 1 with four conic singularities on tori and spheres, focusing on symmetric cases with specific angles, and provides general results along with detailed analysis of particular symmetric metrics.

Contribution

It introduces new results on metrics with four conic singularities, especially symmetric configurations with angles multiple of pi/2, expanding understanding of such geometric structures.

Findings

Characterization of conformal metrics with four conic singularities on tori and spheres.

Detailed analysis of symmetric metrics with angles (pi/2,3pi/2,pi/2,3pi/2).

General results on the structure of these metrics.

Abstract

We discuss conformal metrics of curvature 1 on tori and on the sphere, with four conic singularities whose angles are multiples of pi/2. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles (pi/2,3pi/2,pi/2,3pi/2).