Nash不等式与黎曼流形

Nash Inequality and Riemannian Manifold

运用光滑截断函数的性质,证明了对任一n维完备的黎曼流形,若它的Ricci曲率非负,且满足一个Nash不等式,则它微分同胚于Rn。另外,利用迭代的方法,得到了在没有曲率假设下,若黎曼流形满足Nash不等式,则测地球的体积具有极大增长。

In this paper, we use the property of the smooth cut-off function to prove the following result: for any n-dimensional complete Riemannian manifold with nonnegative Ricci curvature, if one of the Nash inequalities is satisfied, then it is diffeomorphic to Rn . We also use the iterating method to obtain that if the Nash inequalities are satisfied on the Riemannian manifold without any curvature assumption, then the geodesic ball has maximal volume growth.