Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus

TL;DR

This paper establishes a new asymptotically optimal lower bound for the first eigenvalues of large genus Riemann surfaces, linking it to the shortest separating multi-curve length.

Contribution

It introduces a novel lower bound for the first eigenvalue based on geometric properties, proven to be optimal as genus increases.

Findings

First eigenvalue exceeds a constant times the shortest separating multi-curve length divided by g^2.

The lower bound is proven to be asymptotically optimal for large genus.

Provides a geometric-analytic link between eigenvalues and surface topology.

Abstract

In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{\mathcal{L}_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $\mathcal{L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g \to \infty$.