Remarks on $L^{p}$-vanishing results in geometric analysis

TL;DR

This paper surveys $L^{p}$-vanishing results for solutions to geometric equations, introduces new aspects, and discusses applications including a structure theorem for stable minimal hypersurfaces.

Contribution

It provides an overview of $L^{p}$-vanishing results with refined inequalities and presents a new abstract structure theorem for stable minimal hypersurfaces.

Findings

New geometric applications are discussed.

An abstract structure theorem for stable minimal hypersurfaces is introduced.

Refined Kato inequalities are used in the analysis.

Abstract

We survey some $L^{p}$-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schr\"{o}dinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed.