Laplace eigenvalues of ellipsoids obtained as analytic perturbations of the unit sphere

Abstract

The Euclidean unit sphere in dimension n minimizes the first positive eigenvalue of the Laplacian among all the compact, Riemannian manifolds of dimension n with Ricci curvature bounded below by \(n-1\) as a consequence of Lichnerowicz’s theorem. The eigenspectrum of the Laplacian is given by a non-decreasing sequence of real numbers tending to infinity. In dimension two, we prove that such an inequality holds for the subsequent eigenvalues in the sequence for ellipsoids that are obtained as analytic perturbations of the Euclidean unit sphere for the truncated spectrum.