Reconcile The Mapped Boundary And Surface Normalization Of Ricci Flow,

This report consists of two parts.The first part studies the boundary regularity of harmonic maps from hyperbolic space into nonpositively curved Riemannian manifolds. This is the boundary regularity problem of certain uniformly degenerate elliptic operators. Recently, similar problem arise in the study of conformally compact Einstein manifolds and is solved successfully. We will apply this method to the study of boundary regularity of harmonic maps. We prove that near the boundary the harmonic map has an expansion involving log terms and this class of harmonic maps have the best possible regularity. In the proof, adapted to our problem, the original method is simplified.The second part studies normalized Ricci flow on noncompact surfaces. The main theorem says under certain conditions normalized Ricci flow on nonparabolic surfaces will converge to constant curvature metric. The main idea of the proof is to generalize a classical method of Hamilton. For this, we need to overcome some technical difficulty for analysis on noncompact manifolds. In particular, when Ricci curvature is bounded from below, a growth estimate of Green's function is proved. This estimate is true for any dimension and has independent interests. Finally, we apply the result and method to prove the Uniformization theorem in certain cases.