Hitting estimates on Einstein manifolds and applications
-
Beomjun Choi
and
Robert Haslhofer
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the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for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 nonnegative 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.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: 2022R1C1C1013511
Funding statement: The first author has been partially supported by National Research Foundation of Korea grant No. 2022R1C1C1013511, POSTECH Basic Science Research Institute grant No. 2021R1A6A1A10042944, and POSCO Science Fellowship. The second author has been supported by an NSERC Discovery Grant and a Sloan Research Fellowship.
Acknowledgements
The first author thanks the University of Toronto, his affiliation at the time this research was commenced.
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Articles in the same Issue
- Frontmatter
- Kähler–Einstein metrics with prescribed singularities on Fano manifolds
- Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy
- Hermitian K-theory via oriented Gorenstein algebras
- Components of symmetric wide-matrix varieties
- Tangent curves to degenerating hypersurfaces
- Local noncollapsing for complex Monge–Ampère equations
- Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry
- Hitting estimates on Einstein manifolds and applications
- Special values of L-functions of one-motives over function fields
Articles in the same Issue
- Frontmatter
- Kähler–Einstein metrics with prescribed singularities on Fano manifolds
- Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy
- Hermitian K-theory via oriented Gorenstein algebras
- Components of symmetric wide-matrix varieties
- Tangent curves to degenerating hypersurfaces
- Local noncollapsing for complex Monge–Ampère equations
- Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry
- Hitting estimates on Einstein manifolds and applications
- Special values of L-functions of one-motives over function fields
