Annals of Global Analysis and Geometry

, Volume 24, Issue 4, pp 337–349

Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum'

  • Bruno Colbois
  • Ahmad El Soufi
Article

DOI: 10.1023/A:1026257431539

Cite this article as:
Colbois, B. & El Soufi, A. Annals of Global Analysis and Geometry (2003) 24: 337. doi:10.1023/A:1026257431539

Abstract

Let M be a compact connected manifold of dimension n endowed witha conformal class C ofRiemannian metrics of volume one. For any integer k ≥ 0, we consider the conformal invariant λkc(C) defined as the supremum of the k-th eigenvalue λk(g) of the Laplace–Beltrami operator Δg, where g runs over C.

First, we give a sharp universal lower bound for λkc(C) extending to all k a result obtained by Friedlander andNadirashvili for k = 1. Then, we show that the sequence λkc(C), that we call `conformal spectrum',is strictly increasing and satisfies, ∀ k ≥ 0,λk+1c(C)n/2kc(C)n/2nn/2 ωn, where ωn is the volume of the n-dimensionalstandard sphere.

When M is an orientable surface of genus γ, we also considerthe supremum λktop(γ)of λk(g) over theset of all the area one Riemannian metrics on M, and study thebehavior of λktop(γ) in terms of γ.

Laplacian eigenvalue conformal metric universal lower bound 

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Bruno Colbois
    • 1
  • Ahmad El Soufi
    • 2
  1. 1.Laboratoire de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland
  2. 2.Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083Université de Tours, Parc de GrandmontToursFrance

Personalised recommendations