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A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Francisco J. Carreras ,
Fernando Giménez  and
Vicente Miquel
Francisco J. Carreras
Affiliation:
Dept. de Geometría y Topología, Universidad de Valencia, 46100 Burjasot (Valencia), Spain
Fernando Giménez
Affiliation:
Dept. de Mathemática Aplicada, E.T.S.I Industriales, Universidad Politécnica de Valencia, Valencia, Spain
Vicente Miquel
Affiliation:
Dept. de Geometría y Topología, Universidad de Valencia, 46100 Burjasot (Valencia), Spain
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Abstract

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We give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

MSC classification

Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53C55: Hermitian and Kählerian manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 2 , April 1994 , pp. 267 - 277
Copyright
Copyright © Australian Mathematical Society 1994

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