# The sphere covering inequality and its dual

## TL;DR

This paper introduces a new proof of the sphere covering inequality, presents a dual inequality, and extends these results to surfaces with general isoperimetric properties, with applications to elliptic equations in two dimensions.

## Contribution

It provides a novel proof and a dual version of the sphere covering inequality, extending its applicability to broader geometric contexts and elliptic PDEs.

## Key findings

- New proof of the sphere covering inequality
- Discovery of a dual sphere covering inequality
- Applications to elliptic equations with exponential nonlinearities

## 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.