Inserted: 27 feb 2018
Last Updated: 27 feb 2018
We derive a singular version of the Sphere Covering Inequality which was recently introduced in 42, 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 56. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.
Keywords: uniqueness results, Geometric PDEs, Mean field equation, Sphere covering inequality, Singular Liouville-type equations, Alexandrov-Bol inequality