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arXiv · 2512.22598

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

Jianquan Ge·Ya Tao
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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.

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Jianquan Ge, Ya Tao. 2025-12-27. Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature. https://arxiv.org/abs/2512.22598

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