Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds
Author(s)
Date issued
January 21, 2010
In
Bull. Lond. Math. Soc.
Vol
1
No
42
From page
96
To page
108
Subjects
Laplacian eigenvalue upper bound submanifold
Abstract
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.
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.
Publication type
journal article