Hyperbolic Geometric Flow-review And Ideas

This paper is a review. It briefly presents hyperbolic geometric flow intro-duced and studied by the De-Xing Kong and Ke-Feng Liu. Finally, some ideas about hyperbolic geometric flow are given. The thesis is organized as follows.Chapter 1 introduces the concept and some basic properties of hyperbolic geometric flow which make us basically understand that hyperbolic geometry flow is a powerful tool to study some important problems in differential geometry and relativity.Chapter 2 mainly introduces some properties of hyperbolic geometry flow with the dissipate term and some theories of hyperbolic Ricci soliton. We can de-rive wave character of curvature, short-time existence and uniqueness, nonlinear stability, the existence and non-existence of whole classics solutions of dissipative hyperbolic geometric flow on Riemann surfaces.Chapter 3 investigates exact solutions of hyperbolic geometric flow. Through analysing these exact solutions, we derive some relevant conclusions which can help us to understand basic properties of hyperbolic geometry flow and also can help us to understand general Einstein's equation.In Chapter 4, we firstly introduce some basic properties of hyperbolic mean curvature flow, then study hyperbolic mean curvature flow for the evolution of plane curve. By means of the support functions, a hyperbolic Monge—Ampere equation is derived. Based on this, we prove solution's local existences and uniqueness. Furthermore, we investigate dissipative hyperbolic mean curvature flow. Finally, we investigate the formation of singularities in the motion of plane curves under hyperbolic mean curvature flow, some blow-up results have been obtained and estimates on the life-span of the solutions are given.Chapter 5 gives some of open problems about hyperbolic geometric flow.