Equiboundedness of the Weil-Petersson metric

TL;DR

This paper establishes uniform bounds for derivatives of solutions to the negative curvature equation and the Weil-Petersson metric, analyzing their dependence on Riemann surface geometry and providing bounds related to systoles and pinching directions.

Contribution

It introduces new uniform bounds for derivatives of solutions and the Weil-Petersson metric, linking these bounds to geometric features of Riemann surfaces.

Findings

Uniform bounds for derivatives of solutions to the curvature equation.

Bounds for Weil-Petersson curvature tensor derivatives in terms of systoles.

Analysis of norm comparisons for harmonic Beltrami differentials.

Abstract

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichm\"{u}ller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,\alpha}$ and $L^2$ norms for harmonic Beltrami differentials are analyzed. Uniform bounds are given for the covariant derivatives of the Weil-Petersson curvature tensor in terms of the systoles of the underlying Riemann surfaces and the projections of the differentiation directions onto {\it pinching directions}. The main analysis combines Schauder and potential theory estimates with the analytic implicit function theorem.