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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

References

[1]Bang, S. J., ‘The first Dirichlet eigenvalue and radius of a geodesic ball’, Proc. Amer. Math. Soc. 104 (1988), 885–886.Google Scholar
[2]Chavel, I., Eigenvalues in Riemannian geometry (Academic Press, New York, 1984).Google Scholar
[3]Cheng, S. Y., Li, P. and Yau, S. T., ‘Heat equations on minimal submanifolds and their applications’, Amer. J. Math. 106 (1984), 1033–1065.Google Scholar
[4]Giménez, F., ‘Comparison theorems for the volume of a complex submanifold of a Kaehler manifold’, Israel J. Math. 71 (1990), 239–255.CrossRefGoogle Scholar
[5]Giménez, F. and Miquel, V., ‘Bounds for the first Dirichlet eigenvalue of domains in kaehler manifolds’, Archiv der Mathematik 56 (1991), 370–375.Google Scholar
[6]Gray, A., Tubes (Addison-Wesley, New York, Reading, 1990).Google Scholar
[7]Greene, R. E. and Wu, H., Function theory on manifolds which possess a pole, Lecture in Mathematics 699 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[8]Jorge, L. and Koutroufiotis, D., ‘An estimate for the curvature of bounded submanifolds’, Amer. J. Math. 103 (1981), 711–725.Google Scholar
[9]Karcher, H., ‘Riemannian comparison constructions’, in: Global differential geometry (ed. Chern, S. S.) (Math. Assoc. Amer., Washington, D.C., 1989) pp. 170–222.Google Scholar
[10]Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, volume II (Interscience, New York, 1969).Google Scholar
[11]Kowalski, O. and Vanhecke, L., ‘A new formula for the shape operator of a geodesic sphere and its applications’, Math. Zeit. 192 (1986), 613–625.Google Scholar
[12]Markvorsen, S. M., ‘On the heat kernel comparison theorems for minimal submanifolds’, Proc. Amer. Math. Soc. 97 (1986), 479–482.CrossRefGoogle Scholar
[13]Royden, H. L., ‘Comparison theorems for the matrix Riccati equation’, Comm. Pure Appl. Math. 41 (1988), 739–746.Google Scholar