Holomorphic quadratic differentials dual to Fenchel-Nielsen coordinates

TL;DR

This paper constructs dual bases of holomorphic quadratic differentials to Fenchel-Nielsen coordinates, providing foundational estimates crucial for sharp eigenvalue bounds on degenerating hyperbolic surfaces.

Contribution

It introduces a natural basis dual to Fenchel-Nielsen differentials and derives precise estimates essential for eigenvalue analysis on degenerating surfaces.

Findings

Construction of dual bases for holomorphic quadratic differentials

Derivation of precise estimates for degenerating hyperbolic surfaces

Foundation for sharp eigenvalue bounds in related work

Abstract

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in the current paper form the basis for the proof of the sharp eigenvalue estimates on degenerating surfaces obtained in arXiv:1701.08491.