New results on some dynamical and stationary problems in geometry

he results obtained in this dissertation can be regarded as the first step toward our project of studying the following two fields: Einstein manifold theory and harmonic mapping theory.;Inspired by R. Hamilton's work, we begin to study the Einstein manifolds with boundary by introducing suitable boundary value problems for Hamilton's Ricci-flow equations. We prove a short time existence theorem for the Neuman boundary value problem. We also obtain partial results for the Dirichlet boundary value problem. As an application of our result on the Neuman problem, we prove that: any three-dimensional Riemannian manifold with totally geodesic boundary and with positive Ricci curvature is diffeomorphic to a Riemannian manifold with positive constant curvature and with totally geodesic boundary. We also derive a Simons-type identity for hypersurfaces evolving under the Ricci-flow.;Motivated by our studies on harmonic mappings between complete manifolds, we prove the following Liouville theorem: If ;The last two chapters are devoted to some miscellaneous problems that we have studied in the past. We prove two results on rigidity problems for manifolds with boundary in Chapter 4: (1) Let (M,g) be a conformally flat Riemannian manifold with weakly umbilical boundary. If (M,g) has constant scalar curvature and positive Ricci curvature, then (M, g) has constant positive curvature. (2) If a compact manifold (M,g) with weakly umbilical boundary has positive curvature operator and satisfies ;In Chapter 5, we prove a result on a conjecture of Fischer-Marsden: If a Riemannian metric g on a 3-manifold...