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 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 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 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 Biharmonic submanifolds in manifolds with bounded curvature

2016 ◽  
Vol 27 (11) ◽  
pp. 1650089
Author(s):  
Shun Maeta
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Biharmonic Submanifolds ◽  
Bounded Curvature ◽  
Negative Constant ◽  
The Mean ◽  
Biharmonic Submanifold

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

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 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 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.

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