Small eigenvalues of surfaces of finite type

TL;DR

This paper extends previous research on eigenvalues of surfaces, demonstrating that finite type Riemannian surfaces with negative Euler characteristic have a bounded number of small eigenvalues, related to their topological complexity.

Contribution

It establishes a new upper bound on the number of small eigenvalues for finite type surfaces, generalizing prior results to a broader class of surfaces.

Findings

Finite type surfaces with negative Euler characteristic have at most as many small eigenvalues as the absolute value of their Euler characteristic.

The result generalizes earlier work on closed surfaces to surfaces of finite type.

The bound relates spectral properties directly to topological invariants.

Abstract

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and negative Euler characteristic has at most negative of the Euler characteristics many small eigenvalues.