We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.
This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4（-1）,whose scalar curvature is bounded from below.
XU Hong-wei1 ZHAO En-tao2 Center of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.
Leng Yan Xu Hongwei Zhejiang University, Center of Mathematical Sciences
The geometric properties for Gaussian image of submanifolds in a sphere are investigated. The computation formula, geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sphere are obtained.
This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.
A geometric rigidity theorem for submanifolds with parallel mean curvature and positive curvature in a space form is proved. It is a generalization of the famous rigidity theorems due to S. T. Yau and others.
