The Blow-up Of The Conformal Mean Curvature Flow

In this paper,we introduce and study the conformal mean curvature flow of subman-ifolds of higher codimension in the Euclidean space Rn.This kind of flow is a special case of some more general mean curvature flow which is of various origination.As the main result,we prove a blow-up theorem as follows:Theorem 0.1[see Theorem 2.2]Let M be a compact manifold of dimension m ? 2 and a? C?(Rn×[0,T0))be a positive function.Then for any given F0?F(M),there exists a maximal and finite T>0 such that the CMCF(1.6)has a unique maximal solution F:M ×[0,T)?Rn which blows up at the time T in the sense that lim maxM |h|2 = +?,where h.?ht is the second fundamental form of the immersion Ft:M?Rn.Furthermore,by using the idea of Andrews and Baker for studying the mean curva-ture flow of submanifolds in the Euclidean space,we also compute some evolution formulas and inequalities which we believe to be useful in our further study of conformal mean cur-vature flow.Presently,these computations,together with our main theorem,are applied to provide a direct proof of the following convergence theorem:Theorem 0.2[see Theorem 3.2]Let M be as in Theorem 1.1.Suppose that the initial immersion F0?F(M)satisfies the following two conditions:(1)the mean curvature H does not vanish everywhere;(2)the square of the norm of the second fundamental form |h|2?c|H|2 for some constant c satisfying c?4/3m in case 2 ? m ? 4;c?1/m-1 in case m ? 5.(2)Then the mean curvature flow(1.7)or(1.8)with an external conformal force has a unique smooth solution F:M ×[0,T)? Rm+p on a finitte maximal time interval,and Ft(M)converges uniformly to a round point in Rm+p.