scholarly journals Some Integral Formulas for the (r+ 1)th Mean Curvature of a Closed Hypersurface

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Akram Mohammadpouri ◽  
S. M. B. Kashani
Keyword(s):  
Mean Curvature ◽  
Integral Formulas ◽  
Closed Hypersurface

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

2011 ◽  
Vol 54 (1) ◽  
pp. 67-75 ◽  
Author(s):  
QIN ZHANG
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface ◽  
Small Constant

AbstractLet Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

scholarly journals Some Integral Formulas for Hyper-Surfaces in Euclidean Spaces

1971 ◽  
Vol 43 ◽  
pp. 117-125 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Euclidean Spaces ◽  
Integral Formulas

Let M be an oriented hypersurface differentiably immersed in a Euclidean space of n + 1 ≥: 3 dimensions. The r-th mean curvature Kr of M at the point P of M is defined by the following equation:

SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN UNIT SPHERES

2011 ◽  
Vol 22 (01) ◽  
pp. 131-143 ◽  
Author(s):  
GANGYI CHEN ◽  
HAIZHONG LI
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Positive Constant ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface

Let M be an n-dimensional closed hypersurface with constant mean curvature H in a unit sphere Sn+1, n ≤ 8, and S the squared length of the second fundamental form of M. If |H| ≤ ε(n), then there exists a positive constant α(n, H), which depends only on n and H, such that if S0 ≤ S ≤ S0 + α(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ε(n) is a positive constant depending only on n and [Formula: see text].

scholarly journals GRADIENT AND HESSIAN ESTIMATES FOR DIRICHLET AND NEUMANN EIGENFUNCTIONS

2020 ◽  
Vol 71 (1) ◽  
pp. 379-394
Author(s):  
Feng-Yu Wang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Integral Formulas ◽  
Manifold With Boundary ◽  
Hessian Estimates

Abstract We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

scholarly journals Compact space-like hypersurfaces in de Sitter space

2005 ◽  
Vol 2005 (13) ◽  
pp. 2053-2069 ◽  
Author(s):  
Jinchi Lv
Keyword(s):  
Compact Space ◽  
Mean Curvature ◽  
De Sitter Space ◽  
Higher Order ◽  
De Sitter ◽  
Totally Umbilical ◽  
Higher Order Mean Curvatures ◽  
Integral Formulas

We present some integral formulas for compact space-like hypersurfaces in de Sitter space and some equivalent characterizations for totally umbilical compact space-like hypersurfaces in de Sitter space in terms of mean curvature and higher-order mean curvatures.

Integral formulas for the Laplacian along the unstable foliation and applications to rigidity problems for manifolds of negative curvature

1991 ◽  
Vol 11 (4) ◽  
pp. 803-819 ◽  
Author(s):  
Chengbo Yue
Keyword(s):  
Stochastic Process ◽  
Simple Proof ◽  
Geodesic Flow ◽  
Mean Curvature ◽  
Negative Curvature ◽  
Constant Mean Curvature ◽  
Compact Manifolds ◽  
Unstable Foliation ◽  
Integral Formulas ◽  
Rigidity Problems

AbstractWe obtain a class of integral formulas for the Lapacian along unstable leaves of the geodesic flow of compact manifolds of negative curvature. Using these formulas, we give two functional descriptions of those manifolds with horospheres having constant mean curvature. More rigidity problems are discussed, including a simple proof of two important Lemmas by Hamenstadt which avoids her use of stochastic process.

scholarly journals Integral Formulas for Submanifolds and their Applications

1970 ◽  
Vol 22 (2) ◽  
pp. 376-388 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Integral Formulas ◽  
Closed Hypersurfaces

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].

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