Sharp eigenvalue estimates on degenerating surfaces
Abstract
We consider the first non-zero eigenvalue of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that 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 of Albin, Rochon and Sher.
Cite
@article{arxiv.1701.08491,
title = {Sharp eigenvalue estimates on degenerating surfaces},
author = {Nadine Große and Melanie Rupflin},
journal= {arXiv preprint arXiv:1701.08491},
year = {2019}
}
Comments
Major changes to prove sharpness of the result now also for genus 2 surfaces and to split the original paper into two parts for easier publication, with this paper containing the results on eigenvalues, while the used results on holomorphic quadratic differentials are now proven in a separate paper