J Austral Math Soc Ser A 56 pp267--277, 1994.

A Comparison Theorem for the First Dirichlet Eigenvalue of a Domain in a Kaehler Submanifold

Francisco J. Carreras, Fernando Gimenez and Vicente Miquel

Abstract

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.

1991 AMS Subject Classification: primary 58G20; secondary 58G30, 53C21, 53C55

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Authors

Francisco J. Carreras

Dept. de Geometria y Topologia, Universidad de Valencia, 46100 Burjasot (Valencia), Spain.

Fernando Gimenez

Dept. de Mathematica Aplicada, E.T.S.I. Industries, Universidad Politecnica de Valencia, Valencia, Spain.

Vicente Miquel

Dept. de Geometria y Topologia, Universidad de Valencia, 46100 Burjasot (Valencia), Spain.