Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications

TL;DR

This paper extends Hardy-Littlewood-Sobolev inequalities to compact Riemannian manifolds and applies these results to solve a generalized Yamabe problem using a novel variational method.

Contribution

It introduces an extension of Hardy-Littlewood-Sobolev inequalities on compact manifolds and offers a new, simpler approach to solving the Yamabe problem.

Findings

Extended inequalities to dimension n≠2 on compact manifolds

Solved a generalized Yamabe problem on locally conformally flat manifolds

Provided a new variational approach for the Yamabe problem

Abstract

In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy functional and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.