The sphere covering inequality and its dual

Abstract

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

arXiv comment21 pages

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• Existence results for super-Liouville equations on the sphere via bifurcation theory
• Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions
• A remark on the concentration compactness principle in critical dimension
• Improved Moser-Trudinger-Onofri inequality under constraints
• Remarks on a mean field equation on ${\mathbb{S}}^2$
• Entropy maximization in the two-dimensional Euler equations
• Improved Beckner's inequality for axially symmetric functions on ${\mathbb{S}}^n$
• Existence results for a non-relativistic Chern-Simons model with purely mutual interaction
• Existence results for a generalized mean field equation on a closed Riemann surface