Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

Given a manifold M , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the fir...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 51; no. 3-4; pp. 701 - 724
Main Authors Freitas, Pedro, Mao, Jing, Salavessa, Isabel
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2014
Springer Nature B.V
Subjects
Analysis
Calculus of variations
Calculus of Variations and Optimal Control; Optimization
Construction
Control
Curvature
Dirichlet problem
Disks
Eigenvalues
Geodetics
Manifolds
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theorems
Theoretical
35P15
58C40
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Summary:Given a manifold M , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0692-7