# Sharp eigenvalue estimates on degenerating surfaces

## TL;DR

This paper provides sharp estimates for the first non-zero Laplacian eigenvalue on degenerating hyperbolic surfaces, linking it to Fenchel-Nielsen coordinates and refining previous asymptotic results.

## Contribution

It establishes a precise relationship between the eigenvalue's gradient and Fenchel-Nielsen coordinates, improving error estimates and expanding understanding of eigenvalue asymptotics on degenerating surfaces.

## Key findings

- Eigenvalue gradient aligns with Fenchel-Nielsen length coordinate dual
- Improved error rates in eigenvalue estimates
- New insights into eigenvalue expansion terms

## Abstract

We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8\pi \nabla\log(\lambda_1)$ essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve previous results of Schoen, Wolpert, Yau and Burger to obtain estimates with optimal error rates and obtain new information on the leading order terms of the polyhomogeneous expansion of $\lambda_1$ of Albin, Rochon and Sher.