Journal ArticleDOI
The scalar curvature of minimal hypersurfaces in a unit sphere
Young Jin Suh,Hae Young Yang +1 more
TLDR
In this paper, the authors studied n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and gave an answer for S. Chern's conjecture.Abstract:
In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if , where S denotes the squared norm of the second fundamental form of this hypersurface.read more
Citations
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Journal ArticleDOI
Linear weingarten hypersurfaces in a unit sphere
TL;DR: In this paper, the authors considered linear Weingarten hyper-surface in a sphere and obtained some rigidity theorems, which were extended by Cheng-Yau (3) and Li (7).
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On Chernʼs problem for rigidity of minimal hypersurfaces in the spheres
Qi Ding,Yuanlong Xin +1 more
TL;DR: For a compact minimal hypersurface M in S n + 1 with the squared length of the second fundamental form S, it was shown in this paper that there exists a positive constant δ ( n ) depending only on n, such that if n ⩽ S ⩾ n + δ( n ), then S ≡ n, i.e., M is a Clifford minimal hypersuroface, in particular, when n ≥ 6, the pinching constant ω(n ) = n 23.
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On Chern's conjecture for minimal hypersurfaces and rigidity of self-shrinkers☆
Hongwei Xu,Hongwei Xu,Zhiyuan Xu +2 more
TL;DR: In this paper, the generalized Yau parameter method and the Sylvester theory were used to verify that if M is a compact minimal hypersurface in S n + 1 whose squared length of the second fundamental form satisfies 0 ≤ | A | 2 − n ≤ n 22, then | A| 2 ≡ n and M is Clifford torus.
Journal ArticleDOI
The second pinching theorem for hypersurfaces with constant mean curvature in a sphere
Hongwei Xu,Zhiyuan Xu +1 more
TL;DR: In this article, the authors generalized the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of small constant mean curvature.
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A New Pinching Theorem for Closed Hypersurfaces with Constant Mean Curvature in $S^{n+1}$
Hongwei Xu,Ling Tian +1 more
TL;DR: In this article, the generalized Chern conjecture was investigated in a closed hypersurface with constant scalar curvature and constant mean curvature, and it was shown that there exists an explicit positive constant (C(n)$ depending only on n$ such that if H = n, β(n,H) = n + β(H) + 4(n − 1)H^2 + n − 1), then H > \beta (n, H) + 3n − 7 + n−1 + β (n−2) + σ(
References
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Seminar on differential geometry
TL;DR: A wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics, is presented in this paper, which includes a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincare inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and
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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.
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The scalar curvature of minimal hypersurfaces in spheres
Chia-Kuei Peng,Chuu-Lian Terng +1 more
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Chern's conjecture on minimal hypersurfaces
Hongcang Yang,Qing-Ming Cheng +1 more
TL;DR: In this article, it was shown that an n-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with f3 constant is isometric to the totally geodesic sphere or the Clifford torus if S ≤ 1.8252n−0.712898.