On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere

Jinchuan Bai; Yong Luo

On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere

Differential Geometry 2023-02-20 v2

Authors: Jinchuan Bai , Yong Luo

Abstract

Let $M^n$ be an $n$ -dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$ . We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v \right) ^ { \frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.

Keywords

complete hypersurface constant scalar curvature unit sphere locally conformally flat total curvature totally geodesic

Cite

@article{arxiv.2205.13156,
  title  = {On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere},
  author = {Jinchuan Bai and Yong Luo},
  journal= {arXiv preprint arXiv:2205.13156},
  year   = {2023}
}

Comments

Typos and inaccuracies corrected, accepted by Kodai Mathematical Journal

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