Ricci iteration on homogeneous spaces
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- by Artem Pulemotov and Yanir A. Rubinstein PDF
- Trans. Amer. Math. Soc. 371 (2019), 6257-6287
Abstract:
The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov–Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric. In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing. Our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient Ricci flow exists on a given Riemannian manifold as well as a method for predicting its limit.References
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Additional Information
- Artem Pulemotov
- Affiliation: School of Mathematics and Statistics, The University of Queensland, St. Lucia, Queensland, 4072 Australia
- MR Author ID: 696644
- Email: a.pulemotov@uq.edu.au
- Yanir A. Rubinstein
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 795645
- Email: yanir@umd.edu
- Received by editor(s): August 24, 2016
- Received by editor(s) in revised form: September 21, 2017
- Published electronically: January 24, 2019
- Additional Notes: Research of A. P. was supported by the Australian Research Council Discovery Early-Career Researcher Award DE150101548.
Research of Y. A. R. was supported by NSF grants DMS-1206284,1515703 and a Sloan Research Fellowship.
Part of this work took place while Y. A. R. visited MSRI (supported by NSF grant DMS-1440140) during the Spring 2016 semester. - © Copyright 2019 Artem Pulemotov and Yanir A. Rubinstein
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6257-6287
- MSC (2010): Primary 39A12, 53C30, 53C44
- DOI: https://doi.org/10.1090/tran/7498
- MathSciNet review: 3937324
Dedicated: To the memory of Itzhak Bar-Lewaw Mulstock and Serhiy Ovsiyenko