Unboundedness of the first eigenvalue of the laplacian in the symplectic category

Lev Buhovsky

doi:10.1142/S1793525313500040

Unboundedness of the first eigenvalue of the laplacian in the symplectic category

Lev Buhovsky

School of Mathematical Sciences

Tel Aviv University

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

Abstract

Given a closed symplectic manifold (M²ⁿ, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ₁ of the Laplacian associated with it. We show that λ₁ can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.

Original language	English
Pages (from-to)	13-56
Number of pages	44
Journal	Journal of Topology and Analysis
Volume	5
Issue number	1
DOIs	https://doi.org/10.1142/S1793525313500040
State	Published - Mar 2013

Keywords

Laplacian
Riemannian metric
first eigenvalue
quasi-Kähler structure
symplectic manifold

All Science Journal Classification (ASJC) codes

Analysis
Geometry and Topology

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10.1142/S1793525313500040

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title = "Unboundedness of the first eigenvalue of the laplacian in the symplectic category",

abstract = "Given a closed symplectic manifold (M2n, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ1 of the Laplacian associated with it. We show that λ1 can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.",

keywords = "Laplacian, Riemannian metric, first eigenvalue, quasi-K{\"a}hler structure, symplectic manifold",

author = "Lev Buhovsky",

note = "Funding Information: I thank Michael Entov, Dan Mangoubi, Leonid Polterovich and Felix Schlenk for their encouragement, interest in this work, and helpful comments and suggestions. I thank Yaron Ostrover for pointing my attention to Conjecture 1.8, Alexander Bykhovsky and Sobhan Seyfaddini for improving the style of the paper, and the referee for useful suggestions which helped to reduce the size of the paper. The author was partially supported by the NSF Grant DMS-1105813.",

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doi = "10.1142/S1793525313500040",

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N1 - Funding Information: I thank Michael Entov, Dan Mangoubi, Leonid Polterovich and Felix Schlenk for their encouragement, interest in this work, and helpful comments and suggestions. I thank Yaron Ostrover for pointing my attention to Conjecture 1.8, Alexander Bykhovsky and Sobhan Seyfaddini for improving the style of the paper, and the referee for useful suggestions which helped to reduce the size of the paper. The author was partially supported by the NSF Grant DMS-1105813.

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N2 - Given a closed symplectic manifold (M2n, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ1 of the Laplacian associated with it. We show that λ1 can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.

AB - Given a closed symplectic manifold (M2n, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ1 of the Laplacian associated with it. We show that λ1 can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.

KW - Laplacian

KW - Riemannian metric

KW - first eigenvalue

KW - quasi-Kähler structure

KW - symplectic manifold

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DO - 10.1142/S1793525313500040

M3 - مقالة

SN - 1793-5253

VL - 5

SP - 13

EP - 56

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

IS - 1

ER -

Unboundedness of the first eigenvalue of the laplacian in the symplectic category

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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