基于 R. Finn, P. Li 等人的研究， S. Muller 和 Sverak 研究了 Rn 中的浸入二维子流形的第二基本形式和拓扑之间的联系。他们得出，当第二基本形式的积分有某个固定上界时，浸入实际上是嵌入。作者本人继续研究了 R5 中的超曲面，并得出部分结果。作者发现超曲面的总平均曲率有一个固定上界时，该超曲面只有一个 end. 结合 Chang-Qing-Yang 的关于共形平坦四维流形的工作，推广 S. Muller 和 Sverak 的结果有了希望。进一步，很自然地就是去研究高维和余高维子流形是否也能有类似结论。所以，在本项目中，我们将研究子流形的拓扑与几何之间的联系。特别地，利用平均曲率研究什么时候浸入变成嵌入。

Based on the research of R. Finn, P. Li, etc., S. Muller and Sverak studied the relationship between the second fundamental form and topology of an immersed two dimensional manifold in Rn. He obtained that if the total integral of the second fundamental form of the immersed 2-manifold in Rn is bounded by a universal constant, then the immersion is actually embedding. The auther continued to study.the immersed 4-manifold in R5, and have found partial result. The auther found that given an universal bound of the total mean curvature, the immersed hypersurfaces in Rn has only one end. Combining the work of Chang-Qing-Yang on conformally flat 4-manifolds, it shows some light on generalization of S. Muller and Sverak's work. Furthermore, it is natural to study the higher dimension and.codimension submanifold. Hence, in this project, we will study the relationship between the topology and the geometry of submanifolds. Especially, using the mean curvature to find when the immersion is an embedding.