Sharp eigenvalue estimates on degenerating surfaces

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.

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

References in corpus (3)

• Analytic torsion and R-torsion of Witt representations on manifolds with cusps
• Smooth long-time existence of Harmonic Ricci Flow on surfaces
• Resolution of the canonical fiber metrics for a Lefschetz fibration