The Study Of Cauchy Problem For Hyperbolic Mean Curvature Flow

Hyperbolic mean curvature flow is a preliminarily study of hyperbolic partial differential equation theory in differential geometry,biomedicine and physics.It was widely used in the following aspects:simulated soap bubble behavior,crystal evolution,biocytology,the theory of relativistic string(or membrane)in Minkowski space and so on.In this paper,we studied the lifespan of classical solution to Cauchy problem for the hyperbolic mean curvature and the formation of singularities in the motion of spacelike curves in Minkowski R~1,1,1 under generalized hyperbolic mean curvature flow.In chapter 1,we investigated the lifespan of classical solution to Cauchy problem for the hyperbolic mean curvature flow.By means of support function,a hyperbolic Monge-Ampère equation is derived and this equation could be reduced to a system in Riemann invariant.Using the theory of the local solution of Cauchy problem for quasilinear hyperbolic system,we discussed the lifespan(the maximal existence time of unique local classical solutions)of classical solution to the Cauchy problem for the hyperbolic mean curvature flow.In chapter 2,we investigated the formation of singularities in the motion of spacelike curves in Minkowski R~1,1,1 under generalized hyperbolic mean curvature flow.We drived a second-order quasilinear wave equation,and by constructing the Riemann invariants we reduced the wave equation to a reducible quasilinear hyperbolic system of first order.Base on derived quasilinear hyperbolic system,we investigated the formation of singularities in the motion of spacelike curves.In particular,under the generalized hyperbolic mean curvature flow,we proved that the motion of periodic spacelike curves with small variation on one period and small initial velocity blows up in finite time.Some blowup results have been obtained and the estimates on the life-span of the solutions are given.