Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview

Abstract

This note is meant to introduce the reader to a duality principle for nonlinear equations recently discovered in [74,54,50]. 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. Applications, also involving the geometry of submanifolds, will be discussed in the last sections. We conclude by investigating the stability of these maximum principles when we remove polar sets.

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

“…The limit in the first line is equivalent to saying that ̺ is an exhaustion, namely, that it has relatively compact sublevel sets. Its construction relies on a duality principle recently discovered in [38,36,37], called the Ahlfors-Khas'minskii duality (AK-duality for short). Roughly speaking, the principle establishes that, for a large class of operators F including linear, quasilinear and fully nonlinear ones of geometric interest, a Liouville property for solutions u of F [u] ≥ 0 that satisfy sup M u < ∞ is equivalent to the existence of exhaustions w satisfying F [w] ≤ 0.…”

“…Consider the ball B ε centered at p in the metric h, with ε small enough that B 4ε is regular. By the AK-duality [38,36,37], since (N, h) is stochastically complete we can find a function…”

Recent rigidity results for graphs with prescribed mean curvature

“…For bounded from above, the necessity to deduce ( * ) ≤ 0 even if * is not attained motivated the introduction of the fundamental Omori-Yau maximum principles, [103,131,30], that generated an active area of research related to potential theory on manifolds [93,94], with applications to a variety of geometric problems. We refer the reader to [3,10,106] for a detailed account and an extensive set of references.…”

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

“…The existence of easily implies the forward completeness of . The construction of proceeds, as in [37,38], by stacking solutions of obstacle problems, and has independent interest. Implication 3) ⇒ 1) is shown by means of a sequence { } of solutions of Δ ∞ = ( ) defined on an increasing family of relatively compact sets Ω , locally converging to a limit solution ∞ on ∖ , with a small compact set.…”

“…Our investigation arose in the context of fully nonlinear potential theory, motivated by the desire recast, in a unified framework, various maximum principles at infinity available in the literature: the celebrated Ekeland [24,25] and Omori-Yau ones [45,57,18], as well as those coming from stochastic geometry (the weak maximum principles of Pigola-Rigoli-Setti [48], related to parabolicity, stochastic and martingale completeness of a Riemannian manifold). This investigation initiated in [37,38], in a Riemannian setting, see also previous results in [50,49]. The need to consider first order conditions in the statements of Ekeland and Omori-Yau principles requires to include the eikonal into the class of equations to which the theory be applicable, and opened the way to also encompass the ∞-Laplace operator, tightly related to the eikonal one.…”