子流形的具有位置向量方向外力场的平均曲率流

Mean Curvature Flow with a Forcing Field in Direction of the Position Vector of Submanifolds

考虑欧氏空间R^(n+p)中n(≥2)维闭子流形沿平均曲率向量场加上一个位置向量方向外力场的流的发展.设子流形任意一点处平均曲率向量非零和第二基本形式的模长以平均曲率向量长度的常数倍(仅与n有关)为界,我们证明了若外力场很小时,拼挤子流形要么在有限时间内收缩为一点,要么子流形在任意时刻都存在;若外力场足够大时,子流形发散到无穷大;同时,当流发展到极限位置时,规范化子流形都光滑收敛到R^(n+p)中的一个n+1维子空间中的标准球面.

This paper considers the evolution by mean curvature vector plus a forcing field in the direction of its position vector of a closed submanifold of dimension n （〉 2） in I1~n＋p. Suppose that mean curvature vector is nonzero everywhere and that the full norm of the second fundamental form is bounded by a fixed multiple （depending only on n） of the length of the mean curvature vector at every point. It is shown that such submanifolds may contract to a point in finite time if the forcing field is small, or exist for all time and expand to infinity if it is large enough. Moreover, if the evolving submanifolds undergo suitable homotheties and the time parameter is transformed appropriately into a parameter t 〈 ce, it is also shown the normalized submanifolds in any case converge smoothly to a round sphere in some （n ＋ 1）-dimensional subspace of Rn＋p as t → oo.