A uniform Poincar\'e estimate for quadratic differentials on closed surfaces

TL;DR

This paper establishes a universal estimate for quadratic differentials on closed Riemann surfaces, linking their deviation from holomorphic differentials to their antiholomorphic derivatives, applicable across all surfaces of genus at least two.

Contribution

It provides a uniform Poincaré estimate for quadratic differentials on all closed surfaces of genus at least two, extending previous localized results.

Findings

The estimate is valid uniformly for all such surfaces.

It bounds the distance to holomorphic differentials using antiholomorphic derivatives.

The result applies to surfaces of any genus ≥ 2.

Abstract

We prove a uniform estimate, valid for every closed Riemann surface of genus at least two, that bounds the distance of any quadratic differential to the finite dimensional space of holomorphic quadratic differentials in terms of its antiholomorphic derivative.