Geometric and topological rigidity for compact submanifolds of odd dimension

Xu, Hongwei; Leng, Yan; Gu, Juan-Ru

doi:10.1007/s11425-013-4755-1

Cited by 5 publications

(4 citation statements)

References 39 publications

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“…One can verify that Ric(e 1 ∧ e 2 ∧ • • • ∧ e p ) is well defined, i.e., it is depending only on the p-plane e 1 ∧ e 2 ∧ • • • ∧ e p . With an obvious modification of original results of Gu-Xu [17] and Gu-Leng-Xu [18] , one can obtain the following Theorem (for readers' convenience, we list a proof in Section 4). Theorem D. [17;18] Let M be an n(≥ 4)-dimensional closed submanifold with mean curvature vector H in F n+m (c) with c ≥ 0.…”

Section: Moreover By the Definition Of Imentioning

confidence: 99%

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Chen

Sun

2019

Journal of Differential Equations

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In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifoldM for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) ofM bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results and a homology sphere theorem. Second, when the pull back Weitzenböck operator ofM bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates and homology sphere theorems are optimal whenM has constant curvature.2010 Mathematics Subject Classification. 58J50; 53C24; 53C40.

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Section: Moreover By the Definition Of Imentioning

confidence: 99%

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Chen

Sun

2019

Journal of Differential Equations

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“…Remark 1. If h α ijk = 0 for any α, i, j, k in Equation (22), then |T| = 0 or |η| = 0 (i.e., M n is either contained in a slice S m (c), or ∂ t is tangent to R everywhere). In particular, suppose M n is a compact submanifold of S m (c) × R. It is easy to see that |T| = 0, and thus M n lies in S m (c).…”

Section: Preliminariesmentioning

confidence: 99%

“…He proved in 1987 that if M n is a compact PMC submanifold in S n+p (c) with n ≥ 4, and under the additional assumption that the Ricci curvature of M n is not less than n(n − 2)(c + H 2 )/(n − 1), then M n is a totally umbilic sphere. By replacing Sun's pinching constant with the optimal possible constant (n − 2)(c + H 2 ), He and Luo [20] and Xu et al [21,22] further generalized Ejiri's pinching theorem to compact PMC submanifolds in a space form.…”

Section: Introductionmentioning

confidence: 99%

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Qiu,

Zhan

2023

Mathematics

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Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

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“…It is natural to ask that if we can improve the pinching condition. In odd-dimensional case, the pinching constant can be lowered down (see Li [9], Xu-Leng-Gu [17]'s results). In this paper, we will consider the integral Ricci curvature condition instead of the pointwise Ricci curvature condition.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of minimal submanifolds in space forms

Chen,

Wei

2018

Preprint

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In this paper, we consider the rigidity for an n(≥ 4)-dimensional submanfolds M n with parallel mean curvature in the space form M n+p c when the integral Ricci curvature of M has some bound. We prove that, if c + H 2 > 0 and RicHere H is the norm of the parallel mean curvature of M , and ǫ(n, c, λ, H) is a positive constant depending only on n, c, λ and H. This extends some of the earlier work of [15] from pointwise Ricci curvature lower bound to inetgral Ricci curvature lower bound.

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Geometric and topological rigidity for compact submanifolds of odd dimension

Cited by 5 publications

References 39 publications

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in Sm(c)×R

Rigidity of minimal submanifolds in space forms

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