Some Problems Of Curvature Flow

The content of this paper is divided into two chapters. In the first chapter, we study the metric deformation on Riemannian manifolds with boundary. In the second chapter, we investgate the Type II singularity of mean curvature flow of compact hypersurface in Riemannian manifold.It is known to all, the study of the Ricci flow began with Hamilton's seminal 1982 paper 'Three manifolds with positive Ricci curvature.' In that paper he not only introduce the notation of the Ricci flow, but applied it to classify closed 3-manifolds with positive Ricci curvature. Later, in another very important 1986 paper 'Four-manifolds with positive curvature operator,' Hamiton extende his methods to show that closed 4-manifolds with positive curvature operator are topologically either S~4 or RP~4. For the Ricci flow on n-dimensional (n ≥ 4) manifolds, if the initial metric possess positive curvature operator and strong pinching conditions, then we can get the similiar results, consult the reference papers [Hul], [Ma] and [Ni]. In 1995, Hamilton [Ha3] studied the behavior of the singularity of Ricci flow. The study of the Ricci flow on complete noncompact Riemannian manifolds began with Shi's paper [Shi1] and [Shi2]. B. L. Chen and X. P. Zhu in [CZ] consider the Ricci flow on complete Riemannian manifolds and get a Bonnet-Myers type result. Recently, Perelman' paper [P1] and [P2] are seen as a move towards settling the Poincaré conjecture finally. In 1996, Shen [Shen] applied Hamilton's Ricci flow to study the metric deformation on Riemannian manifolds with boundary. Shen prove a short time existence theorem for manifolds with umbilical boundary. He also derived the Simons' identity for the boundary under the Ricci flow. And as a corollary, Shen show that any three-manifolds with totally geodesic boundary which admits positive Ricci curvature can be deformed to a space form with totally geodesic...