The Mean Convex Hypersurface In Hyperbolic Space

The mean curvature flow is one of the most popular research fields in differential geometry in recent years. Let M0be an n-dimensional smooth manifold and let F0M0nâ†'Nn+r be a smooth immersion. We want to study the evolution of F(·, t) by mean curvature flow; that is, the family of immersions F(·, t) satisfying where H(·, t) and v(·, t) are the mean curvature and the outer normal respectively at the point F(p, t) of the surface Mt=F(·, t)(M). What is we cared that with the develop-ment of the time, the tendency of the surface.In the previous we have studied the mean curvature flow in Euclidean spaces. Let n≥2, r=1and M0is uniformly, then the evolution equation (0.2) has a smooth solution on a finite time interval and the Mt’s converge to a single point as tâ†'T. From now on we consider the mean convex hypersurface in hyperbolic space, and that the flow develops a singularity of Type II as tâ†'T. Then the limit hypersurface F∞(·, Ï„) is weakly convex.