Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric

A. M. Micheletti; Angela Pistoia

doi:10.3934/dcds.1998.4.709

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1998, Volume 4, Issue 4: 709-720. Doi: 10.3934/dcds.1998.4.709

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Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric

A. M. Micheletti^1, and
Angela Pistoia^2,

1.
Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa
2.
Dip. di Mat. Appl. "U. Dini", Università di Pisa

Received: September 30, 1996

Revised: March 31, 1998

Published: September 1998

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Abstract

Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$

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Mathematics Subject Classification: 58G18, 58G25.

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