New Estimates for the First Eigenvalue of the Jacobi Operator on Closed Hypersurfaces in Riemannian Space Forms

被引：0

作者：

Liu, Jiancheng ^{[1
]}

Mi, Rong ^{[1
]}

机构：

[1] Northwest Normal Univ, Coll Math & Stat, 967 Anning E Rd, Lanzhou 730070, Gansu, Peoples R China

来源：

CZECHOSLOVAK MATHEMATICAL JOURNAL | 2020年 / 70卷 / 03期

基金：

中国国家自然科学基金;

关键词：

Jacobi operator; first eigenvalue; closed hypersurface; CONSTANT MEAN-CURVATURE; STABILITY EIGENVALUE; RIGIDITY THEOREMS;

D O I：

10.21136/CMJ.2020.0579-18

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study the first eigenvalue of the Jacobi operator on closed hypersurfaces with constant mean curvature in non-flat Riemannian space forms. Under an appropriate constraint on the totally umbilical tensor of the hypersurfaces and following Melendez's ideas in J. Melendez (2014) we obtain a new sharp upper bound of the first eigenvalue of the Jacobi operator.

引用

页码：881 / 890

页数：10

共 23 条

[1] HYPERSURFACES WITH CONSTANT MEAN-CURVATURE IN SPHERES [J].

ALENCAR, H ;

DOCARMO, M .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1994, 120 (04) :1223-1229

[2] A spectral characterization of the H(r)-torus by the first stability eigenvalue [J].

Alías, LJ ;

Barros, A ;

Brasil, A .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2005, 133 (03) :875-884

[3] Hypersurfaces with constant scalar curvature in space forms [J].

Alias, Luis J. ;

Melendez, Josue ;

Palmas, Oscar .

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 2018, 58 :65-82

[4] An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms [J].

Alias, Luis J. ;

Garcia-Martinez, S. Carolina .

GEOMETRIAE DEDICATA, 2012, 156 (01) :31-47

[5]

Alías Luis J., 2006, Rev. Unión Mat. Argent., V47, P39

[6] ON THE FIRST STABILITY EIGENVALUE OF HYPERSURFACES IN THE EUCLIDEAN AND HYPERBOLIC SPACES [J].

Aquino, Cicero P. ;

de Lima, Henrique F. ;

dos Santos, Fabio R. ;

Velasquez, Marco A. L. .

QUAESTIONES MATHEMATICAE, 2017, 40 (05) :605-616

[7] STABILITY OF HYPERSURFACES OF CONSTANT MEAN-CURVATURE IN RIEMANNIAN-MANIFOLDS [J].

BARBOSA, JL ;

DOCARMO, M ;

ESCHENBURG, J .

MATHEMATISCHE ZEITSCHRIFT, 1988, 197 (01) :123-138

[8]

Chavel I., 1984, Eigenvalues in Riemannian geometry

[9] Estimates for the first eigenvalue of Jacobi operator on hypersurfaces with constant mean curvature in spheres [J].

Chen, Daguang ;

Cheng, Qing-Ming .

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2017, 56 (02)

[10] EIGENVALUE COMPARISON THEOREMS AND ITS GEOMETRIC APPLICATIONS [J].

CHEN, SY .

MATHEMATISCHE ZEITSCHRIFT, 1975, 143 (03) :289-297

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