Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

Abstract

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.

67 pages. Final version

References in corpus (5)

• On the Distributional Hessian of the Distance Function
• Global maximum principles and divergence theorems on complete manifolds with boundary
• A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation
• Characterizing the Strong Maximum Principle
• On viscosity solutions to the Dirichlet problem for elliptic branches of nonhomogeneous fully nonlinear equations

Cited by in corpus (8)

• Bernstein and half-space properties for minimal graphs under Ricci lower bounds
• Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian
• Recent rigidity results for graphs with prescribed mean curvature
• Non-negative Ricci curvature and Minimal graphs with linear growth
• A barrier principle at infinity for varifolds with bounded mean curvature
• A splitting theorem for capillary graphs under Ricci lower bounds
• Stochastic half-space theorems for minimal surfaces and $H$-surfaces of ${\mathbb{R}}^3$
• On splitting complete manifolds via infinity harmonic functions