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

By B. Colbois, Ahmad El Soufi. 2003

101 citations recorded by the source corpus.

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/2− kc(C)n/2≥ nn/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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb001fa9af8715bd6de35988e4763f1ba8e2cc0e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Maximization of the second positive Neumann eigenvalue for planar domains (2008, background): “It was proved in [ CE , Corollary 1] that (1.2.2) λck(S n ,[g0]) := sup”
A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle (2006, background): “ What are the extremal metrics on M? (2) Is the supremum Λ(M) achieved and, if so, by what extremal metrics? (3) How does Λ(M) depend on (the genus of) M? Concerningthe last question, it follows from [6] that Λ(M) is an increasing function of the genus with a linear growth rate. Explicit answers to ”
An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “Gluing these spheres together using suﬃciently thin and short tubes yields a metric g on the sphere with ¯ λ k ( S 2 , g ) close to µ k (see [9, Section 2] for details).”
Laplacian eigenvalue functionals and metric deformations on compact manifolds (2007, background): “In [ 5 ], Colbois and the first author proved that”