Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces

TL;DR

This paper extends curvature estimates and sheeting theorems to weakly stable constant mean curvature hypersurfaces, broadening understanding beyond strongly stable cases and providing effective compactness results.

Contribution

It establishes new pointwise curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces, generalizing classical results for strongly stable hypersurfaces.

Findings

Established a pointwise curvature estimate in non-singular dimensions.

Proved a sheeting theorem applicable in all dimensions.

Provided an effective version of the compactness theorem for weakly stable CMC hypersurfaces.

Abstract

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving an effective version of the compactness theorem for weakly stable CMC hypersurfaces established in the recent work of the first and third-named authors. Our results generalize the curvature estimate and the sheeting theorem proven respectively by Schoen--Simon--Yau and Schoen--Simon for strongly stable hypersurfaces.