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.