The Pohozaev-Schoen Identity on Asymptotically Euclidean Manifolds: Conservation Identities and their applications

TL;DR

This paper extends the Pohozaev-Schoen identity to asymptotically Euclidean manifolds, providing new conservation identities and applications such as rigidity results for Ricci-solitons and static potentials.

Contribution

It introduces a generalized Pohozaev-Schoen identity for asymptotically Euclidean manifolds and demonstrates its applications in geometric analysis and rigidity theorems.

Findings

Rigidity results for asymptotically Euclidean Ricci-solitons

An almost-Schur inequality without Ricci curvature restrictions

Rigidity results for static potentials from conservation principles

Abstract

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur-type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.