| In Riemannian geometry,it is important to ask if there is a conformal change of the initial metric that makes the scalar curvature to be constant.That is the Yamabe problem.This problem was solved by Yamabe,Trudinger,Aubin and Schoen in 1980s.A different approach to the Yamabe problem is the Yamabe flow,finally by use of the general concentration-compactness result of Schwetlick and Struwe,Brendle proved the convergence of the flow.The CR geometry,which is the abstract model of real hypersurfaces in complex manifolds,have a lot of analogy with the geometry of Riemannian manifolds.We can also consider a Yamabe type problem on CR manifolds.To distinguish it with the Yamabe problem,it is called the CR Yamabe problem.Also,we can introduce the CR Yamabe flow.And in this paper,We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold M of real dimension 2n + 1.We prove a concentration-compactness theorem when n = 1,and use a CR type positive mass theorem to prove the convergence of the CR Yamabe flow when n = 1 or M is locally CR equivalent to the CR sphere S2n+1.And also for completeness,we prove the exponential convergence of the CR Yamabe flow when CR Yamabe constant is zero.At last,we introduce a geometric flow called area-preserving mean curvature flow which has free Neumann boundaries.We show that for a rotationally symmetric n-dimensional hypersurface in Rn+1 between two parallel hyperplanes will converge to a cylinder with the same area under this flow.We use the geometric proper-ties and the maximal principle to obtain gradient and curvature estimates,leading to long-time existence of the flow and convergence to a constant mean curvature surface. |
