Some remarks on the Pigola-Rigoli-Setti version of the Omori-Yau maximum principle

TL;DR

This paper establishes that the Pigola-Rigoli-Setti version of the Omori-Yau maximum principle is equivalent to the existence of a proper $C^2$ function with bounded gradient and Hessian on the manifold, broadening its applicability.

Contribution

It proves the logical equivalence between the Pigola-Rigoli-Setti hypotheses and the existence of a specific proper function, extending the Omori-Yau maximum principle beyond curvature bounds.

Findings

Equivalence of hypotheses and proper function existence

Extension of Omori-Yau principle scope

Broader applicability beyond curvature bounds

Abstract

We prove that the hypotheses in the version of the Omori-Yau maximum principle that was given by Pigola-Rigoli-Setti are logically equivalent to the assumption that the manifold carries a $C^2$ proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori-Yau principle, formulated in terms of lower bounds for curvature.