A steady length function for Ricci flows
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- by Joshua Jordan PDF
- Proc. Amer. Math. Soc. 149 (2021), 397-406 Request permission
Abstract:
A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons. A similar quantity was found by Feldman, Ilmanen, and Ni [J. Geom. Anal. 15 (2005), pp. 49–62] which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.References
- Michael Feldman, Tom Ilmanen, and Lei Ni, Entropy and reduced distance for Ricci expanders, J. Geom. Anal. 15 (2005), no. 1, 49–62. MR 2132265, DOI 10.1007/BF02921858
- Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
- Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, ArXiv e-prints, (Feb. 2008). arXiv:math/0211159
Additional Information
- Joshua Jordan
- Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697-3875
- ORCID: 0000-0001-6968-4672
- Email: jpjorda1@uci.edu
- Received by editor(s): April 4, 2020
- Received by editor(s) in revised form: May 25, 2020
- Published electronically: October 16, 2020
- Communicated by: Jia-Ping Wang
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 397-406
- MSC (2020): Primary 53E20
- DOI: https://doi.org/10.1090/proc/15202
- MathSciNet review: 4172614