Stable hypersurfaces with constant scalar curvature in Euclidean spaces

TL;DR

This paper proves nonexistence results for certain stable hypersurfaces with constant scalar curvature in Euclidean spaces, highlighting conditions under which such hypersurfaces cannot exist.

Contribution

It establishes new nonexistence theorems for complete noncompact stable hypersurfaces with nonnegative constant scalar curvature in Euclidean spaces, including a specific case in A4.

Findings

No complete noncompact strongly stable hypersurface in A4 with zero scalar curvature, nonzero Gauss-Kronecker curvature, and finite total curvature.

Nonexistence results for stable hypersurfaces with constant scalar curvature in Euclidean spaces.

Specific nonexistence in A4 for hypersurfaces with certain curvature conditions.

Abstract

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).