Rigidity theorems for hypersurfaces with constant scalar curvature in a unit sphere
Guoxin Wei,Young Jin Suh +1 more
TLDR
In this paper, Cheng and Yau gave a characterization of Clifford tori and in a unit sphere S n + 1 (1) (1), where n is the number of vertices in the sphere.Abstract:
In this paper, we give a characterization of Clifford tori and in a unit sphere S n +1 (1). Our results extend the results due to Cheng and Yau [ 4 ], and Wang and Xia [ 11 ].read more
Citations
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Journal ArticleDOI
A maximum principle for hypersurfaces with constant scalar curvature and applications
TL;DR: In this article, a weak maximum principle for complete hypersurfaces with constant scalar curvature into Riemannian space forms was established and applied to estimate the norm of the traceless part of its second fundamental form.
Journal ArticleDOI
Hypersurfaces with constant scalar curvature in space forms
TL;DR: In this article, the authors studied the rigidity of complete hypersurfaces with constant scalar curvature in Riemannian space forms and proved that either the hypersurface is totally umbilical or it holds a sharp estimate for the supremum of the norm of Φ, with equality if and only if the hyperssurface is isoparametric with two distinct principal curvatures.
Journal ArticleDOI
Complete weingarten hypersurfaces satisfying an okumura type inequality
TL;DR: In this paper, the authors consider complete linear Weingarten hypersurfaces immersed into Riemannian space forms and prove that either the hypersurface is totally umbilical or holds an estimate for the norm of the total umbilicity tensor.
Hypersurfaces with constant scalar or mean curvature in a unit sphere
Shichang Shu,Annie Yi Han +1 more
TL;DR: In this article, it was shown that M is isometric to the Riemannian product S 1 (p 1 i c 2 ) £ S ni1 (c), c 2 = ni2 nr if r ‚ ni2 ni1
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A gap theorem for constant scalar curvature hypersurfaces
TL;DR: In this paper, the authors obtained a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints.
References
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Book ChapterDOI
Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
TL;DR: In this article, an n-dimensional manifold which is minimally immersed in a unit sphere of dimension n+p is considered. But the manifold is not a sphere, it is a manifold.
Journal ArticleDOI
Hypersurfaces with constant scalar curvature
Shiu-Yuen Cheng,Shing-Tung Yau +1 more
TL;DR: The classification of complete 2D surfaces with constant curvatures in R 3 was studied in this paper. But the classification of 2D complete surfaces with non-zero curvatures was not studied.
Journal ArticleDOI
Minimal Hypersurfaces in a Riemannian Manifold of Constant Curvature
TL;DR: In this paper, the authors investigated the converse problem for minimal hypersurfaces in Sn+1 and showed that if the number of principal curvatures is two and the multiplicities of them are at least two for a hypersurface of this kind in Sn+, then it is known that it is possible to construct a minimal hypergraph with a constant number of curvatures.