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Small eigenvalues of surfaces of finite type

Part of: Global differential geometry Spectral theory and eigenvalue problems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  05 June 2017

Werner Ballmann ,
Henrik Matthiesen  and
Sugata Mondal
Werner Ballmann
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany email [email protected]
Henrik Matthiesen
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany email [email protected]
Sugata Mondal
Affiliation:
Department of Mathematics, Indiana University, Rawles Hall, 831 E 3rd Street, Bloomington, IN 47405, USA email [email protected]
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Abstract

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface $S$ of finite type and Euler characteristic $\unicode[STIX]{x1D712}(S)<0$ has at most $-\unicode[STIX]{x1D712}(S)$ small eigenvalues.

Keywords

Laplace operatorsmall eigenvaluesEuler characteristic

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 53C99: None of the above, but in this section 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1747 - 1768
Copyright
© The Authors 2017 

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