Harmonic maps and the geometry of Teichmuller space
Doctor of Philosophy thesis
In this thesis work, we investigate the asymptotic behavior of the sectional curvatures of the Weil-Petersson metric on Teichmuller space. It is known that the sectional curvatures are negative. Our method is to investigate harmonic maps from a nearly noded surface to nearby hyperbolic structures, hence to study the Hopf differentials associated to harmonic maps and the analytic formulas resulting from the harmonicity of the maps. Besides providing a quantitative result, our estimates imply that even though the sectional curvatures are negative, they are not staying away from zero. In other words, we show that when the complex dimension of Teichmuller space T is greater than one, then there is no negative upper bound for the sectional curvature of the Weil-Petersson metric. During the proof, we also give the explicit description of a family of tangent planes which are asymptotically flat.