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Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview

Luciano Mari, Leandro Freitas Pessoa

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.

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 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.

Paper Structure

This paper contains 13 sections, 11 theorems, 75 equations.

Table of Contents

  1. Prelude: maximum principles at infinity
  2. An example: immersions into cones
  3. Parabolicity, capacity and Evans potentials
  4. Link with stochastic processes
  5. On weak formulations: the case of quasilinear operators
  6. The general framework
  7. Ahlfors, Khas'minskii properties and the AK-duality
  8. Applications
  9. Completeness, viscosity Ekeland principle and $\infty$-parabolicity
  10. The Hessian principle and martingale completeness
  11. Laplacian principles
  12. Partial Trace (Grassmannian) operators
  13. AK-duality and polar sets

Key Result

Theorem 1.3

maririgoli Let $\varphi : X^m \rightarrow \mathbb{R}^{2m-1}$ be an isometric immersion. Denoting with $\rho$ the distance from a fixed origin $o$, assume that the sectional curvature of $X$ satisfies for some constant $C>0$. Then, $\varphi(X)$ cannot be contained into any non-degenerate cone of $\mathbb{R}^{2m-1}$.

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof : Sketch of the proof:
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7: Strong Laplacian $\neq$ weak Laplacian
  • Remark 1.8: Geometric conditions for weak Laplacian principle
  • Definition 1.9
  • ...and 32 more