Small eigenvalues of the Laplace operator on compact Riemann surfaces

Small Eigenvalues of the Laplace Operator on Compact Riemann Surfaces by Burton Randol

Let Sf be a compact Riemann surface, which we will assume to have curvature normalized to be — 1 , and let 0=A0<A1^A2^• • be the eigenvalues corresponding to the problem AF+AF=0 on Sf \ where A is the Laplacian for £P. In an otherwise very interesting and useful paper [2], McKean has stated that it is always the case that A ^ J. In this paper, we will show that this need not be true, and that i...

Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

We give upper bounds for the eigenvalues of the Laplace-Beltrami operator of a compact m-dimensional submanifold M of R. 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 pplane in a generic position (transverse to M), or an invariant which measures the concentration of the...

Notes on Compact Riemann Surfaces

Eigenvalues of the Laplace Operator on Certain Manifolds.

To every compact Riemannian manifold M there corresponds the sequence 0 = X1 < X2 < X3 .< ... of eigenvalues for the Laplace operator on M. It is not known just how much information about M can be extracted from this sequence.' This note will show that the sequence does not characterize M completely, by exhibiting two 16-dimensional toruses which are distinct as Riemannian manifolds but have th...

The resultant on compact Riemann surfaces

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the ...