Brownian motion on Perelman’s almost Ricci-flat manifold

Esther Cabezas-Rivas; Robert Haslhofer

doi:10.1515/crelle-2019-0014

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Brownian motion on Perelman’s almost Ricci-flat manifold

Esther Cabezas-Rivas
Esther Cabezas-Rivas
Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60325 Frankfurt am Main, Germany

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De Gruyter De Gruyter Brill | Google Scholar
und Robert Haslhofer
Robert Haslhofer
Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON M5S 2E4, Canada

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De Gruyter De Gruyter Brill | Google Scholar

Veröffentlicht/Copyright: 13. Juni 2019

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2020 Heft 764

Abstract

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold $ℳ = M \times 𝕊^{N} \times I$ $ℳ=M×𝕊N×I$ , whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for $N \to \infty$ $N→∞$ converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on $ℳ$ $ℳ$ and for the horizontal Laplacian on the orthonormal frame bundle $𝒪 ℳ$ $𝒪⁢ℳ$ . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

Funding statement: The first named author has been partially supported by the MINECO (Spain) and FEDER project MTM2016-77093-P. The second named author has been partially supported by NSERC grant RGPIN-2016-04331, NSF grant DMS-1406394, a Connaught New Researcher Award, and a Sloan Research Fellowship.

Acknowledgements

We are indebted to Peter Topping for the proposal of applying the results in [20] to Perelman’s manifold in order to recover the corresponding statements in [12], which inspired the present paper. We would like to thank The Fields Institute for Research in Mathematical Sciences for providing an excellent research atmosphere and hosting the Thematic Program on Geometric Analysis from July–December 2017, when this work was completed.

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Received: 2018-01-17

Revised: 2019-03-02

Published Online: 2019-06-13

Published in Print: 2020-07-01

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