[Paper Review] Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview

Luciano Mari, Leandro Freitas Pessoa|arXiv (Cornell University)|Jan 16, 2018

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper establishes a duality between maximum principles at infinity and the existence of Khas'minskii potentials, unifying various geometric maximum principles (e.g., Omori-Yau, Ekeland) via potential-theoretic frameworks. The key contribution is the Ahlfors-Khas'minskii duality, which links the Ahlfors property (a viscosity formulation of maximum principles) with the existence of exhaustion functions (Khas'minskii potentials), with applications to stochastic completeness, parabolicity, and submanifold geometry.

ABSTRACT

This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-Setti), as well as to describe their interplay with properties coming from stochastic analysis on manifolds. The duality involves an appropriate version of these principles formulated for viscosity subsolutions of fully nonlinear inequalities, called the Ahlfors property, and the existence of suitable exhaustion functions called Khas'minskii potentials. We discuss applications, also involving the geometry of submanifolds, in the last sections, as well as the stability of these maximum principles when we remove polar sets.

Motivation & Objective

• To unify disparate maximum principles at infinity—such as Ekeland’s, Omori-Yau, and Laplacian principles—within a single potential-theoretic framework.
• To establish a duality between the Ahlfors property (viscosity formulation of maximum principles) and the existence of Khas'minskii potentials (exhaustion functions with specific growth conditions).
• To investigate the stability of maximum principles under removal of polar sets, particularly in relation to subequations and monotonicity cones.
• To extend the duality to nonlinear and fully nonlinear PDEs, including Hessian and partial trace operators, and to relate them to geometric and stochastic properties of manifolds.
• To provide a systematic framework for analyzing stochastic completeness, parabolicity, and martingale completeness via potential-theoretic duality.

Proposed method

• Formalizes the Ahlfors property as a viscosity condition on subsolutions of fully nonlinear inequalities, generalizing classical maximum principles.
• Introduces Khas'minskii potentials as exhaustion functions satisfying $ w \to \infty $ at infinity and $ \Delta w \leq G(w) $ or $ \nabla^2 w \leq G(w)\langle\,,\,\rangle $, with $ \int^\infty \frac{ds}{G(s)} = \infty $.
• Establishes the AK-duality: the Ahlfors property holds on $ X \setminus \Sigma $ if and only if $ \Sigma $ is $ F $-polar, under assumptions $ (\mathscr{H}1) $ and $ (\mathscr{H}2) $.
• Applies the duality to $ k $-subharmonic functions and monotonicity cones $ M_k $, showing that $ M_k $-polar sets are removable for subequations with $ M_k $ as monotonicity cone.
• Uses the Riesz characteristic $ p_F $ to characterize when a universal subequation $ F $ admits the Ahlfors property, linking it to the geometry of the subequation.
• Analyzes the stability of the Ahlfors property under removal of compact polar sets $ \Sigma $, proving that if $ \Sigma $ is $ F $-polar, then the Ahlfors property persists on $ X \setminus \Sigma $.

Experimental results

Research questions

• RQ1 How can various maximum principles at infinity—such as Ekeland’s, Omori-Yau, and Laplacian principles—be unified under a single potential-theoretic framework?
• RQ2 What is the precise duality between the Ahlfors property (viscosity formulation of maximum principles) and the existence of Khas'minskii potentials?
• RQ3 Under what conditions does the Ahlfors property persist when removing a polar set $ \Sigma $ from the manifold?
• RQ4 How do $ M_k $-polar sets relate to the solvability of fully nonlinear PDEs and the geometry of submanifolds?
• RQ5 What is the role of the Riesz characteristic $ p_F $ in determining whether a subequation admits the Ahlfors property?

Key findings

• The Ahlfors-Khas'minskii duality holds: $ \widetilde{F} $ has the Ahlfors property on $ X \setminus \Sigma $ if and only if $ \Sigma $ is $ F $-polar, under $ (\mathscr{H}1) $ and $ (\mathscr{H}2) $.
• If $ \Sigma $ is $ F $-polar, then the Ahlfors property for $ \widetilde{F} $ is preserved on $ X \setminus \Sigma $, even when $ \Sigma $ is removed.
• For truncated cone subequations, the Ahlfors property on $ X \setminus \Sigma $ is equivalent to $ \Sigma $ being $ F $-polar.
• If $ \Sigma $ is $ M $-polar for a monotonicity cone $ M $ of $ F $, then $ \widetilde{F} $ has the Ahlfors property on $ X \setminus \Sigma $, due to the existence of weak Khas'minskii potentials.
• The duality extends to $ k $-subharmonic functions: $ M_k $-polar sets with locally finite $ (p_F - 2) $-dimensional Hausdorff measure are $ M_k $-polar, and hence the Ahlfors property holds on the complement.
• The Riesz characteristic $ p_F \geq k $ is necessary and sufficient for the cone $ M_k $ to be a monotonicity cone of $ F $, which in turn ensures the existence of appropriate Khas'minskii potentials.