  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.
