A steady length function for Ricci flows
Author:
Joshua Jordan
Journal:
Proc. Amer. Math. Soc. 149 (2021), 397-406
MSC (2020):
Primary 53E20
DOI:
https://doi.org/10.1090/proc/15202
Published electronically:
October 16, 2020
MathSciNet review:
4172614
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Abstract | References | Similar Articles | Additional Information
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.
- 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 https://doi.org/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 https://doi.org/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
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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
Article copyright:
© Copyright 2020
American Mathematical Society