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Transactions of the American Mathematical Society

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Perelman’s entropies for manifolds with conical singularities


Authors: Klaus Kröncke and Boris Vertman
Journal: Trans. Amer. Math. Soc. 374 (2021), 2873-2908
MSC (2020): Primary 53E20; Secondary 53C25, 58C05
DOI: https://doi.org/10.1090/tran/8295
Published electronically: January 21, 2021
MathSciNet review: 4223036
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Abstract: In this paper we discuss Perelman’s $\lambda$-functional, Perelman’s Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.


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Additional Information

Klaus Kröncke
Affiliation: Department of Mathematics, University Hamburg, Germany
MR Author ID: 1093667
ORCID: 0000-0001-7933-0034
Email: klaus.kroencke@uni-hamburg.de

Boris Vertman
Affiliation: Department of Mathematics, Universität Oldenburg, Germany
MR Author ID: 871560
Email: boris.vertman@uni-oldenburg.de

Keywords: Perelman’s entropy, Ricci flow, Ricci solitons, singular spaces
Received by editor(s): April 3, 2019
Received by editor(s) in revised form: August 16, 2020
Published electronically: January 21, 2021
Additional Notes: The authors were partially supported by DFG Priority Programme “Geometry at Infinity”. The authors thank the Priority programme “Geometry at Infinity” of the German Research Foundation for financial and intellectual support.
Article copyright: © Copyright 2021 American Mathematical Society