Stable constant mean curvature hypersurfaces are area minimizing in small L^1 neighborhoods

TL;DR

This paper proves that strictly stable constant-mean-curvature hypersurfaces in dimensions up to 7 are uniquely area minimizing within small L^1 neighborhoods, ensuring their optimality under volume constraints.

Contribution

It establishes the first rigorous proof that such hypersurfaces are area minimizing in small L^1 neighborhoods, extending stability results to a variational setting.

Findings

Strictly stable constant-mean-curvature hypersurfaces are area minimizing in small L^1 neighborhoods.

Uniqueness of homological area minimization for these hypersurfaces.

Results hold in manifolds of dimension up to 7.

Abstract

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.