Hitting estimates on Einstein manifolds and applications
Abstract
We generalize the Benjamini-Pemantle-Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for two independent Brownian motions on manifolds with non-negative Ricci curvature and positive asymptotic volume ratio. We also obtain generalizations of (1) and (2) for the manifolds with two-sided Ricci bounds and Einstein manifolds with nonzero Einstein constant.
Cite
@article{arxiv.2010.15860,
title = {Hitting estimates on Einstein manifolds and applications},
author = {Beomjun Choi and Robert Haslhofer},
journal= {arXiv preprint arXiv:2010.15860},
year = {2021}
}