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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
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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.


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Additional Information

Joshua Jordan
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697-3875
Email: jpjorda1@uci.edu

DOI: https://doi.org/10.1090/proc/15202
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