Martingales arising from minimal submanifolds and other geometric contexts

TL;DR

This paper studies a class of martingales on Cartan-Hadamard manifolds, providing conditions for their transience and convergence, with applications to minimal submanifolds and other geometric structures, using elementary comparison geometry techniques.

Contribution

It extends previous results on the transience and angular convergence of martingales related to minimal submanifolds and introduces new curvature conditions for harmonic functions.

Findings

Martingales on Cartan-Hadamard manifolds can be transient under certain conditions.

Angular components of these martingales converge almost surely in specific curvature settings.

Minimal submanifolds admit non-constant bounded harmonic functions under certain ambient curvature conditions.

Abstract

We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan-Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito's rule.