Colbois, Bruno

2010-1-21, Colbois, Bruno, Dryden, Emily B, El Soufi, Ahmad

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact m-dimensional submanifold M of R^{m+p}. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of M with a p-plane in a generic position (transverse to M), or an invariant which measures the concentration of the volume of M in R^{m+p}. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when p=1), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for m>2) the differential structure.

2008-12-21, Colbois, Bruno, Dryden, Emily B, El Soufi, Ahmad

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S-2 endowed with S-1-invariant metrics, we consider the subsequence lambda(G)(k) of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If. G has dimension at least 1, we show that the functional lambda(G)(k) admits no extremal metric under volume-preserving G-invariant deforma- tions. If, moreover, M has dimension at least three, then the functional lambda(G)(k) is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard 0(n)-action on S-n; however, if we also require the metric to be induced by an embedding of S-n in Rn+1, we get an optimal upper bound on lambda(G)(k).