Calculus of Variations and Geometric Measure Theory

L. Mari - L. F. Pessoa

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

created by mari1 on 13 Dec 2020


Published Paper

Inserted: 13 dec 2020
Last Updated: 13 dec 2020

Journal: Comm. Anal. Geom.
Number: 2
Pages: 395-497
Year: 2020
Doi: 10.4310/CAG.2020.v28.n2.a6


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.