Applications of Laplacian and Hessian Comparison Theorems

Abstract

Rauch 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 also Calabi [10], Cheeger and Gromoll [13, Moreover, making use of the method by Wu, we have shown in The purpose of the present paper is to give several applications of our comparison theorems.