Bounds on Multiplicities of Laplace Operator Eigenvalues on Surfaces

TL;DR

This paper extends bounds on the multiplicities of Laplace eigenvalues from simple surfaces to more complex surfaces with holes and positive genus, broadening the understanding of spectral properties in geometric analysis.

Contribution

It generalizes existing bounds on Laplace eigenvalue multiplicities to include surfaces with holes and positive genus, expanding the scope of spectral geometry results.

Findings

Bounds on eigenvalue multiplicities are extended to complex surfaces.

The results apply to surfaces with positive genus and holes.

The generalization broadens the applicability of spectral bounds.

Abstract

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus with holes.