具有正截面曲率的子流形中的稳定积分流

On Stable Integral Currents in Submanifolds with Positive Sectional Curvature

设Ｍｍ是空间形Ｎｎ（ｃ）中的余维不大于２的紧致子流形，Ｍ的平均曲率向量关于法联络平行且不等于零．本文证明了，如果Ｍ具有正截面曲率，则Ｍ中不存在稳定积分流．所得结果给出了Ｌａｗｓｏｎ－Ｓｉｍｏｎｓ猜想的部分解答．

Let Mm be a compact submanifold immersed, with nonzero parallel mean curvature vector, in a space form Nn(c). In this paper, we shall prove that if the codimension (n-m) of the submanifold M in N is not greater than 2 and M is of positive sectional curvature, then there are no stable integral currents in M and the homology group Hp(M,Z)=0 for any p∈(0,m). The obtained results give a partial answer for the conjecture of H. B. Lawson and J. Simons.