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Stability of spectra of Hodge-de Rham laplacians

Stability of spectra of Hodge-de Rham laplacians Math. Z. 224, 327–345 (1997) c Springer-Verlag 1997 1 2;? Garth A. Baker ,Jozef Dodziuk Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA e-mail: garth@math.utk.edu Ph.D. Program in Mathematics, Graduate School and University Center, City University of New York, New York, NY 10036-8099, USA e-mail: jozek@hodge.gc.cuny.edu Received 17 October 1994; in final form 5 May 1995 1 Introduction In this paper we investigate the stability of eigenspaces of the Laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact, oriented, Riemannian manifold with boundary (this includes, in particular, both Neumann and Dirichlet conditions for the Laplace-Beltrami op- erator on functions). More precisely, our main result is that the gap between corresponding eigenspaces (precise definition will be recalled below) measured using the L norm, converges to zero when smooth metrics g converge to g in 1 0 the C topology. It is quite well known (cf. [3] or [14]) that the eigenvalues of the Laplacian vary continuously under C -continuous perturbations of the met- ric. It is perhaps less well known, but implicit in the work of Cheeger [3], that eigenspaces vary continuously as subspaces of L when the metric is perturbed 0 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Stability of spectra of Hodge-de Rham laplacians

Baker, Garth A.; Dodziuk, Józef
Mathematische Zeitschrift , Volume 224 (3) – Mar 1, 1997
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References (20)

Publisher
Springer Journals
Copyright
Copyright © 1997 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004293
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 224, 327–345 (1997) c Springer-Verlag 1997 1 2;? Garth A. Baker ,Jozef Dodziuk Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA e-mail: garth@math.utk.edu Ph.D. Program in Mathematics, Graduate School and University Center, City University of New York, New York, NY 10036-8099, USA e-mail: jozek@hodge.gc.cuny.edu Received 17 October 1994; in final form 5 May 1995 1 Introduction In this paper we investigate the stability of eigenspaces of the Laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact, oriented, Riemannian manifold with boundary (this includes, in particular, both Neumann and Dirichlet conditions for the Laplace-Beltrami op- erator on functions). More precisely, our main result is that the gap between corresponding eigenspaces (precise definition will be recalled below) measured using the L norm, converges to zero when smooth metrics g converge to g in 1 0 the C topology. It is quite well known (cf. [3] or [14]) that the eigenvalues of the Laplacian vary continuously under C -continuous perturbations of the met- ric. It is perhaps less well known, but implicit in the work of Cheeger [3], that eigenspaces vary continuously as subspaces of L when the metric is perturbed 0

Journal

Mathematische Zeitschrift – Springer Journals

Published: Mar 1, 1997

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