scholarly journals Hopf-type theorem for self-shrinkers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hilário Alencar ◽  
Gregório Silva Neto ◽  
Detang Zhou
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Type Theorem ◽  
Complex Function ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Weighted Mean Curvature ◽  
Radial Weight

Abstract In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

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

scholarly journals Null 2-type Chen surfaces

1995 ◽  
Vol 37 (2) ◽  
pp. 233-242 ◽  
Author(s):  
Shi-Jie Li
Keyword(s):  
Mean Curvature ◽  
Spectral Decomposition ◽  
Harmonic Functions ◽  
Curvature Vector ◽  
Minimal Submanifolds ◽  
Dimensional Euclidean Space ◽  
Position Vector ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Image Position

Let M be an n-dimensional connected submanifold in an mdimensional Euclidean space Em. Denote by δ the Laplacian of M associated with the induced metric. Then the position vector x and the mean curvature vector H of Min Em satisfyThis yields the following fact: a submanifold M in Em is minimal if and only if all coordinate functions of Em, restricted to M, are harmonic functions. In other words, minimal submanifolds in Emare constructed from eigenfunctions of δ with one eigenvalue 0. By using (1. 1), T. Takahashi proved that minimal submanifolds of a hypersphere of Em are constructed from eigenfunctions of δ with one eigenvalue δ (≠0). In [3, 4], Chen initiated the study of submanifolds in Em which are constructed from harmonic functions and eigenfunctions of δ with a nonzero eigenvalue. The position vector x of such a submanifold admits the following simple spectral decomposition:for some non-constant maps x0and xq, where A is a nonzero constant. He simply calls such a submanifold a submanifold of null 2-type.

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 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 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 QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

2015 ◽  
Vol 92 (1) ◽  
pp. 133-144 ◽  
Author(s):  
JULIAN SCHEUER
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Uniformly Convex ◽  
Explicit Dependence ◽  
Closed Hypersurfaces ◽  
Bounded Volume ◽  
The Mean ◽  
Convex Hypersurfaces

We prove${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of${\it\delta}$on${\it\epsilon}$within the class of uniformly convex hypersurfaces with bounded volume.

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].

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