Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature

TL;DR

This paper studies the rigidity of complete hypersurfaces in Euclidean spaces with constant mean and scalar curvature, providing new characterizations and extending known theorems to higher dimensions.

Contribution

It offers new rigidity results and characterizations for hypersurfaces with constant mean and scalar curvature, extending previous theorems to dimensions 4, 5, 6, and higher.

Findings

Rigidity results for hypersurfaces in dimensions 4 and 5

Characterizations under Gaussian-Kronecker curvature conditions

Extension of rigidity theorems to dimension 6 and higher

Abstract

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide characterizations for the unsolved cases of N\'u\~nez's theorems in dimensions 4 and 5, as well as several rigidity results under some conditions of $r$-th mean curvatures. Moreover, for the case of dimension 6, we also present analogous rigidity results. Finally, for general dimensions, we offer a rigidity theorem under similar pinching conditions.