Equiboundedness of the Weil-Petersson metric

Abstract

Given a topological type for surfaces of negative Euler characteristic, uniform bounds are developed for derivatives of solutions of the 2 2 -dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller 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^0 , C 2 , α C^{2,\alpha } and L 2 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 pinching directions. The main analysis combines Schauder and potential theory estimates with the analytic implicit function theorem.