Gap theorems for minimal submanifolds in R
Gap theorems for minimal submanifolds in R
复制标题
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
Lei Ni
中科院分区：
文献类型：
--
作者：
Lei Ni
设m为单位S(1)中维数为m的紧致浸没最小子流形。J. Simons在[S]中证明，如果kAk < m(n m) 2n 2m 1，其中A是第二基本形式，则m是完全测地线。后来S. S. Chern， M. Do Carmo和S. Kobayashi也用[C-D-K]中的运动框架证明了这一点。对于R中的最小子流形，在[Al]中证明了体积增长有一个类似的定理。更准确地说，Allard证明了R的最小子流形ifM具有欧几里得体积增长，且密度函数M(x； R) = vol (M\Br(x)) Vm 0 (1)rm < 1 + Æ，对于某个小正数Æ，则M是完全测地线。另一方面，文献[An]、[F-C]和[Ty]表明，当M是R中的最小子流形（维数为M）时，总标量曲率R M kAk M dv与M的拓扑和莫尔斯指数密切相关。最近在[S-Z]中证明，如果M是一个稳定的最小超曲面，其总标量曲率为零，则M是完全测地线。在这篇简短的笔记中，我们将展示总标量曲率的一些间隙定理。这些可以被认为是上面提到的r中最小子流形的Simons定理的类比。更准确地说，我们可以证明以下结果：（我们应该指出，我们不需要稳定性假设，这在上面提到的Shen-Zhu [S-Z]的结果中是必不可少的。）
LetM be a compact immersed minimal submanifold of dimensionm in the unit S(1). It was proved by J. Simons in [S] that if kAk < m(n m) 2n 2m 1 , where A is the second fundamental form, then M is totally geodesic. It was also proved by S. S. Chern, M. Do Carmo and S. Kobayashi using the moving frame in [C-D-K] later. For the minimal submanifolds in R it was proved in [Al] that there is a similar theorem for the volume growth. More precisely, Allard showed that ifM, a minimal submanifold of R has Euclidean volume growth and the density function M(x; r) = V ol(M\Br(x)) Vm 0 (1)rm < 1 + Æ, for some small positive number Æ, then M is totally geodesic. On the other hand, it was shown in [An], [F-C] and [Ty] that when M is a minimal submanifold (of dimension m)in R, the total scalar curvature R M kAk m dv is closed related to the topology and the Morse index of M . More recently it was shown in [S-Z] that if M is a stable minimal hypersurface with nite total scalar curvature then M is totally geodesic. In this short note we will show that there are some gap theorems for the total scalar curvature. These may be thought as the analogy of the above mentioned Simons' theorem for the minimal submanifolds in R. More precisely we can show the following result: (We should point out that we do not need the stability assumption, which is essential in the above mentioned Shen-Zhu's result in [S-Z].)