Problems Involving Mean Curvature Flow And The Free Boundary Problem On CR Manifolds

In this dissertation,we mainly study two important problems related to minimal surface,one is the mean curvature flow(including curve shortening flow),the other is the free boundary problem.For the former,we study the long-time existence,con-vergence,and singularity.For the latter,we consider the free boundary problem in a 3-dimensional pseudo-Hermitian manifold.We introduce the solution of the free bound-ary problem and discuss the stability.In chapter 1,we suppose that M is a product of compact Riemann surfaces and?(f)is a graph in M of a strictly area dereasing map f.Let(M,g(t),?t(f))evolve along the Kahler-Ricci mean curvature flow.We show that ?t(f)remains to be a graph of a strictly area decreasing map along the flow and exists for all time.In the positive scalar curvature case,we prove the convergence of the flow and the curvature decay along the flow at infinity.In chpater 2,we consider the curve shortening flow in a general Riemannian man-ifold.Altschuler's results about the flow for space curves are generalized.For any n-dimensional(n?2)Riemannian manifold(M,g)with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.Next,we focus on the 3-dimensional pseudo-Hermitian manifold.In chpater 3,we introduce a curve shortening flow in a 3-dimensional pseudo-Hermitian manifold with vanishing torsion.The flow preserves the Legendrian condi-tion and decreases the length of curves.The stationary solution of the flow is a Legen-drian geodesic.We classify the singularity and prove some convergence results.More-over,we study the flow in Heisenberg group especially with Type I singularity.Finally,we introduce the notion of free boundary constant p-mean curvature(CPMC)surface in a 3-dimensional pseudo-Hermitian manifold N with boundary M.It arises as a critical point of the p-area among surfaces which divides N into two sub-sets of preassigned volumes and whose boundary is free to move in M.Assuming that N is torsion free and the boundary curve of a free boundary CPMC surface ? contains no singular points of M and ?,we introduce a stability criterion for the free bound-ary CPMC surface ?.When M is the Pansu sphere S1 in the Heisenberg group H1 and N is the interior of M,we find examples of free boundary CPMC surfaces which are rotationally symmetric about the t-axis.Besides the p-minimal disk {r=0} ? N,there exist Pansu spherical caps which are stable free boundary CPMC surfaces inter-secting S1.We also find free boundary CPMC surfaces intersecting S1 at two circles{t=±t1}? S1,which are nodoid type hypersurface and unduloid type hypersurface.