【作者】 陈旭忠；

【导师】 白正国； 沈一兵；

【作者基本信息】 浙江大学 ， 基础数学， 2006， 博士

【摘要】 本文分成两章。在第一章中，我们讨论了高维带边黎曼流形上的Ricci流。在第二章中我们讨论了一般黎曼流形中紧致超曲面在平均曲率流下的形变并且对它们的第二类奇点进行了分析。 Ricci流的研究始于Hamilton的1982年的文章[Ha1]。在这篇文章Hamilton不仅引入了Ricci流这个概念，并且证明了具有正Ricci曲率的闭3-流形上一定存在着常正曲率度量。接着，在另外一篇非常重要的文章[Ha2]中，Hamilton进一步利用Ricci流的方法证明了任何有着正曲率算子的闭4-流形是拓扑的S~4或RP~4。对于维数n≥4的黎曼流形，如果初始的度量的正曲率算子加上足够强的拼挤条件，也能够得到类似的结果，参见[Hu1]，[Ma]和[Ni]。在1995年，Hamilton在文[Ha3]中研究了Ricci流的奇点。完备非紧黎曼流形上Ricci流的研究则是由Shi在[Shi1]，[Shi2]开始的。进一步的通过考虑完备黎曼流形上的Ricci流，陈兵龙和朱熹平在文[CZ]中还得到了一个判断完备流形一定是紧致流形的Bonnet-Myers型定理。最近在[P1]，[P2]中，Perelman利用Ricci流的方法向Poincaré猜想的最后解决又迈进了一大步。而带边流形上的Ricci流的研究始于Shen[Shen]，在1996年，Shen在[Shen]中考虑带边三维流形上的黎曼度量的Ricci形变，证明了如果初始三维流形的黎曼度量具有正Ricci曲率和全测地边界，则此三维黎曼流形上存在着常正曲率的黎曼度量。更多还原

【Abstract】 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更多还原