Hardy and Rellich Inequalities for submanifolds in Hadamard spaces

TL;DR

This paper extends Hardy and Rellich inequalities to submanifolds within Hadamard spaces, providing new theoretical results, applications, and analysis of equality cases in this geometric context.

Contribution

It proves the general Hardy and Rellich inequalities for submanifolds in Hadamard spaces, advancing the understanding of integral inequalities in differential geometry.

Findings

Established Hardy and Rellich inequalities for submanifolds in Hadamard spaces

Provided applications of these inequalities in geometric analysis

Analyzed conditions for equality cases in the inequalities

Abstract

Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon and Hoffman-Spruck. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. Years later, Carron obtained a Hardy inequality for submanifolds in Hadamard spaces. In this paper, we prove the general Hardy and Rellich Inequalities for submanifolds in Hadamard spaces. Some applications are given and we also analyse the equality cases.