In this paper, we investigate under which conditions a differential inequality F on a manifold X satisfies a Liouville-type property. This question, arising in nonlinear potential theory, is related to the validity of an appropriate maximum principle at infinity for F (called here the Ahlfors property) which for instance includes, for suitable F, the classical Ekeland and Omori-Yau principles, their weak versions in the sense of Pigola-Rigoli-Setti, and properties coming from stochastic geometry. Our main goal is to describe a unifying duality between the Ahlfors property and the existence of suitable exhaustions called Khas'minskii potentials, that applies to a large class of fully-nonlinear operators F of geometric interest, and allow to discover new relations between the principles above. Applications include the investigation of the martingale and geodesic completeness of X, and submanifold geometry.

中文翻译：

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非线性方程的 Ahlfors-Liouville 和 Khas'minskii 性质之间的对偶性

在本文中，我们研究了流形 X 上的微分不等式 F 在哪些条件下满足 Liouville 型性质。这个问题出现在非线性势论中，与 F 的无穷大适当最大值原理（此处称为 Ahlfors 性质）的有效性有关，例如，对于合适的 F，包括经典的 Ekeland 和 Omori-Yau 原理，它们的弱点Pigola-Rigoli-Setti 意义上的版本，以及来自随机几何的性质。我们的主要目标是描述 Ahlfors 性质和称为 Khas'minskii 势的合适穷竭的存在之间的统一对偶性，它适用于一大类几何感兴趣的完全非线性算子 F，并允许发现原理之间的新关系以上。