A Remark on 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 $g:M\to \mathbb{R}$ there is a sequence $x_n\in M$, such that ${\lim }_{n\to \infty }g(x_n)={\sup }_{M}g$, ${\lim }_{n\to \infty }|\nabla g(x_n)|=0$ and ${\mathrm{limsup}}_{n\to \infty }\Delta g(x_n)\le 0$. It is shown that if the Ricci curvature does not approach $-\infty$ too fast the manifold satisfies the Omori-Yau maximum principle. This improves earlier necessary conditions. The given condition is quite optimal.

arXiv commentTo appear in the Kuwait Journal of Science and Engineering

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• On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds
• On the Omori-Yau Maximum Principle and Geometric Applications
• An Omori-Yau maximum principle for semi-elliptic operators and Liouville-type theorems