Applications of Laplacian and Hessian Comparison Theorems
Applications of Laplacian and Hessian Comparison Theorems
复制标题
拉普拉斯和 Hessian 比较定理的应用
DOI：
10.2969/aspm/00310333
复制
发表时间：
1984
影响因子：
3.1
通讯作者：
A. Kasue
中科院分区：
文献类型：
--
作者：
A. Kasue
Rauch [41]证明了关于Jacobi域长度的一个基本定理，称为Rauch比较定理。在他之后，Berger [4]，Warner [48]，Heintze和Karcher [27]等人对Rauch比较定理进行了推广。特别是Heintze和Karcher证明了子流形的正规指数映射的长度和体积畸变的一个非常一般的比较定理。他们的比较定理的证明反过来告诉我们一些有用的信息的“局部”行为的拉普拉斯和海森的距离函数的子流形（参见。格林和吴[25]在子流形是点的情况下）。另一方面，吴[49]证明了，在某些情况下，距离函数的拉普拉斯算子和海森算子在适当的弱意义下可以从上面“全局”估计（参见[10]）。[10]，Cheeger and Gromoll [13，14]，Yau [51]）。此外，利用吴的方法，我们在[32]中证明了距离函数的Laplacian和Hessian的一般比较定理。本文的目的是给出我们的比较定理的几个应用。
Rauch [41] proved a fundamental theorem on the lengths of Jacobi fields, called the Rauch comparison theorem. After him, Berger [4], Warner [48] and Heintze and Karcher [27] etc. had extended the Rauch comparison theorem. Especially, Heintze and Karcher showed a very general comparison theorem for the length and volume distortion of the normal exponential map of a submanifold. The proof of their comparison theorem in turn tells us some useful informations about the "local" behaviour of the Laplacian and Hessian of the distance function to a submanifold (cf. Greene and Wu [25] in the case when a submanifold is a point). On the other hand, Wu [49] has proved that, in certain situations, the Laplacian and the Hessian of a distance function in an appropriate weak sense can be "globally" estimated from above (cf. also Calabi [10], Cheeger and Gromoll [13, 14], Yau [51]). Moreover, making use of the method by Wu, we have shown in [32] general comparison theorems on the Laplacian and the Hessian of a distance function. The purpose of the present paper is to give several applications of our comparison theorems.