The Motion Of Curves Under Hyperbolic Mean Curvature Flow

In this thesis, we introduce a kind of hyperbolic mean curvature flow (HMCF) and study the motion of plane curves under HMCF. We also derive the explicit formulation of the exact solution to the motion equations of relativistic strings in Minkowski space R1+n. In Appendix, we study the physical characters of KLS time-periodic universe. The thesis is organized as follows.In Chapter 1 we will present the background of mean curvature flow (MCF). The central problems under consideration and the main results obtained in this thesis are stated.Chapter 2 introduces the hyperbolic mean curvature flow for the evolution of the plane curves. By means of the support function, a hyperbolic Monge-Ampere equation is derived. Based on this, we prove that there exists a class of initial velocities such that the solution of the hyperbolic mean curvature flow blows up in finite time, and the limit solution shrinks to a point or a continuous but piecewise smooth closed weakly convex curve. We also consider the hyperbolic mean curvature flow with the dissipate term and derive the hyperbolic partial differential equations satisfied by the support function and the mean curvature of the curve, respectively. Furthermore, we investigate the relation between the hyperbolic mean curvature flow and the equations of the motion of relativistic strings in Minkowski space R1,1.Chapter 3 investigates the formation of singularities in the motion of plane curves under hyperbolic mean curvature flow. In the case of one-dimensional graphs, a quasilin-ear system is derived. Based on this, some blow-up results have been obtained and the estimates on the life-span of the solutions are given.In Chapter 4, by the Riemann function of the hyperbolic operator, the explicit formula of the exact solution to the motion of relativistic strings in Minkowski space R1+n. Based on this, we can furthermore investigate the properties of relativistic strings in Minkowski space R1+n. Finally, in Appendix, we study the physical characters of KLS time-periodic universe. By the Penrose diagram of the time-periodic solution, we learn about the topological structure of the whole periodic universe.