Stochastic Completeness and the Omori-Yau Maximum Principle

摘要

A Riemannian manifold M is said to satisfy the Omori-Yau maximum principle if for any C-2 bounded function g : M -> R there is a sequence x(n) is an element of M, such that lim(n ->infinity) g(x(n)) = sup(M) g, lim(n ->infinity) |del g(x(n))| = 0 and lim sup(n ->infinity) Delta g(x(n)) <= 0. On the other hand, M is said to satisfy the Weak-Omori-Yau maximum principle if for any C-2 bounded function g : M -> R there is a sequence x(n) is an element of M, such that lim(n ->infinity) g(x(n)) = sup(M) g and lim sup(n ->infinity) Delta g(x(n)) <= 0. It is easy to construct non-complete examples which are weak-Omori-Yau but not Omori-Yau. In this note, a complete example is constructed.