Stable hypersurfaces with zero scalar curvature in Euclidean space

TL;DR

This paper investigates the stability of four-dimensional Euclidean hypersurfaces with zero scalar curvature, establishing nonexistence results under certain conditions and providing criteria for stability based on curvature bounds.

Contribution

It proves the nonexistence of complete stable hypersurfaces with zero scalar curvature under specific growth and curvature conditions, and offers a stability criterion involving mean and Gauss-Kronecker curvatures.

Findings

No complete stable hypersurface with zero scalar curvature exists under the given conditions.

Provides a sufficient condition for stability based on curvature bounds and extrinsic ball radius.

Establishes a link between curvature properties and stability of hypersurfaces.

Abstract

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.