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Eigenvalue estimates and isoperimetric inequalities for cone-manifolds

Published online by Cambridge University Press:  17 April 2009

Craig Hodgson  and
Johan Tysk
Craig Hodgson
Affiliation:
Department of Mathematics, University of Melbourne, Parkville Vic 3052, Australia, cdh@mundoe.maths.mu.oz.au
Johan Tysk
Affiliation:
Department of Mathematics, Uppsala University Thunbergsvagen, 3 S-75238 Uppsala, Sweden, johant@bellatriz.tdb.uu.se
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Abstract

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This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.

We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.

We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 1 , February 1993 , pp. 127 - 143
Copyright
Copyright © Australian Mathematical Society 1993

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