scholarly journals Hypersurfaces with constant mean curvature and two principal curvatures in n+1

2004 ◽  
Vol 76 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Luis J. Alías ◽  
Sebastião C. de Almeida ◽  
Aldir Brasil Jr.
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Integral Formula ◽  
Second Fundamental Form ◽  
Euclidean Sphere ◽  
Principal Curvatures ◽  
Clifford Torus ◽  
The Second Fundamental Form

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

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 $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

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 Global pinching theorems of submanifolds in spheres

2002 ◽  
Vol 31 (3) ◽  
pp. 183-191
Author(s):  
Kairen Cai
Keyword(s):  
Fundamental Form ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Parallel Mean Curvature Vector ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature ◽  
Set Up

LetMbe a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphereS n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) toLpestimate for the square lengthσof the second fundamental form and the norm of a tensorΦ, related to the second fundamental form, we set up some rigidity theorems. Denote by‖σ‖ptheLpnorm ofσandHthe constant mean curvature ofM. It is shown that there is a constantCdepending only onn,H, andkwhere(n−1) kis the lower bound of Ricci curvature such that if‖σ‖ n/2<C, thenMis a totally umbilic hypersurface in the sphereS n+1.

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A SPHERE

2009 ◽  
Vol 51 (2) ◽  
pp. 413-423 ◽  
Author(s):  
QING-MING CHENG ◽  
YIJUN HE ◽  
HAIZHONG LI
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 M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, 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 Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

A note on rigidity of spacelike self-shrinkers

2020 ◽  
Vol 31 (05) ◽  
pp. 2050035
Author(s):  
Yong Luo ◽  
Hongbing Qiu
Keyword(s):  
Euclidean Space ◽  
Growth Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Integral Method ◽  
Acta Math ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
A Minor

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82].

scholarly journals Self-Expanders of the Mean Curvature Flow

2021 ◽  
Author(s):  
Knut Smoczyk
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Vector ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Mean Convex

AbstractWe study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

scholarly journals Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 244
Author(s):  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Izhar Ahmad ◽  
Akram Ali
Keyword(s):  
Fundamental Form ◽  
Complex Projective Space ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Euclidean Sphere ◽  
Slant Submanifold ◽  
The Second Fundamental Form ◽  
Complex Projective Spaces ◽  
Complex Projective

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

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