First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures

Yingxiang Hu; Hongwei Xu; Entao Zhao

doi:10.1007/s10455-015-9454-4

First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures

Yingxiang Hu
, Hongwei Xu
, Entao Zhao^*

^*Corresponding author for this work

Zhejiang University

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

In this article, we prove pinching theorems for the first eigenvalue $$\lambda _1(M)$$λ1(M) of the Laplacian on compact Euclidean hypersurfaces involving the integrals of $$k$$k-th mean curvature. Particularly, we show that under a suitable pinching condition, the hypersurface is starshaped and almost-isometric to a standard sphere. Based on our theorems, we prove some pinching results for the almost-Einstein and almost-umbilical hypersurfaces in Euclidean space.

Original language	English
Pages (from-to)	23-35
Number of pages	13
Journal	Annals of Global Analysis and Geometry
Volume	48
Issue number	1
DOIs	https://doi.org/10.1007/s10455-015-9454-4
State	Published - 29 Jun 2015
Externally published	Yes

Keywords

Euclidean hypersurfaces
First eigenvalue
differentiable pinching theorem
k-th mean curvature

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10.1007/s10455-015-9454-4

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title = "First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures",

abstract = "In this article, we prove pinching theorems for the first eigenvalue \$\$\textbackslash{}lambda \_1(M)\$\$λ1(M) of the Laplacian on compact Euclidean hypersurfaces involving the integrals of \$\$k\$\$k-th mean curvature. Particularly, we show that under a suitable pinching condition, the hypersurface is starshaped and almost-isometric to a standard sphere. Based on our theorems, we prove some pinching results for the almost-Einstein and almost-umbilical hypersurfaces in Euclidean space.",

keywords = "Euclidean hypersurfaces, First eigenvalue, differentiable pinching theorem, k-th mean curvature",

author = "Yingxiang Hu and Hongwei Xu and Entao Zhao",

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T1 - First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures

AU - Hu, Yingxiang

AU - Xu, Hongwei

AU - Zhao, Entao

PY - 2015/6/29

Y1 - 2015/6/29

N2 - In this article, we prove pinching theorems for the first eigenvalue $$\lambda _1(M)$$λ1(M) of the Laplacian on compact Euclidean hypersurfaces involving the integrals of $$k$$k-th mean curvature. Particularly, we show that under a suitable pinching condition, the hypersurface is starshaped and almost-isometric to a standard sphere. Based on our theorems, we prove some pinching results for the almost-Einstein and almost-umbilical hypersurfaces in Euclidean space.

AB - In this article, we prove pinching theorems for the first eigenvalue $$\lambda _1(M)$$λ1(M) of the Laplacian on compact Euclidean hypersurfaces involving the integrals of $$k$$k-th mean curvature. Particularly, we show that under a suitable pinching condition, the hypersurface is starshaped and almost-isometric to a standard sphere. Based on our theorems, we prove some pinching results for the almost-Einstein and almost-umbilical hypersurfaces in Euclidean space.

KW - Euclidean hypersurfaces

KW - First eigenvalue

KW - differentiable pinching theorem

KW - k-th mean curvature

UR - https://www.scopus.com/pages/publications/84929956007

U2 - 10.1007/s10455-015-9454-4

DO - 10.1007/s10455-015-9454-4

M3 - 文章

AN - SCOPUS:84929956007

SN - 0232-704X

VL - 48

SP - 23

EP - 35

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 1

ER -

First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures

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Keywords

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