Stochastic Completeness and the Omori–Yau Maximum Principle

Abstract

A Riemannian manifold M is said to satisfy the Omori–Yau maximum principle if for any $C^2$ bounded function $g:M\rightarrow {\mathbb {R}}$ there is a sequence $x_n\in M$, such that $\lim _{n\rightarrow \infty }g(x_n)=\sup _M g$, $\lim _{n\rightarrow \infty }|\nabla g(x_n)|=0$ and $\limsup _{n\rightarrow \infty }\Delta g(x_n)\le 0$. On the other hand, M is said to satisfy the Weak-Omori–Yau maximum principle if for any $C^2$ bounded function $g:M\rightarrow {\mathbb {R}}$ there is a sequence $x_n\in M$, such that $\lim _{n\rightarrow \infty }g(x_n)=\sup _M g$ and $\limsup _{n\rightarrow \infty }\Delta g(x_n)\le 0$. 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.