Open Access
Summer 2014 Martingales arising from minimal submanifolds and other geometric contexts
Robert W. Neel
Illinois J. Math. 58(2): 323-357 (Summer 2014). DOI: 10.1215/ijm/1436275486


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.


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Robert W. Neel. "Martingales arising from minimal submanifolds and other geometric contexts." Illinois J. Math. 58 (2) 323 - 357, Summer 2014.


Received: 30 December 2013; Revised: 11 October 2014; Published: Summer 2014
First available in Project Euclid: 7 July 2015

zbMATH: 1322.58023
MathSciNet: MR3367651
Digital Object Identifier: 10.1215/ijm/1436275486

Primary: 58J65
Secondary: 53C42 , 60H30

Rights: Copyright © 2014 University of Illinois at Urbana-Champaign

Vol.58 • No. 2 • Summer 2014
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