A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in Gui and Moradifam (Invent Math. https://doi.org/10.1007/s00222-018-0820-2 , 2018) suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in Luo and Tian (Proc Am Math Soc 116(4):1119–1129, 1992). Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.