Systole and $\lambda_{2g-2}$ of a hyperbolic surface

TL;DR

This paper establishes a geometric lower bound for a specific Laplacian eigenvalue on closed hyperbolic surfaces, linking spectral properties to the surface's systole, using topological methods.

Contribution

It provides an explicit lower bound on the eigenvalue aa_{2g-2}a for hyperbolic surfaces, connecting spectral data with geometric systole measures.

Findings

aa_{2g-2}(S) > 1/4 +
b0(S) for hyperbolic surface S

The lower bound depends explicitly on the systole of S

Topological methods are effective in deriving spectral bounds

Abstract

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${\lambda_{2g-2}}(S)$: ${\lambda_{2g-2}}(S) > 1/4 + {\epsilon_0}(S)$, where ${\epsilon_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$