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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Perelman’s entropies for manifolds with conical singularities
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by Klaus Kröncke and Boris Vertman PDF
Trans. Amer. Math. Soc. 374 (2021), 2873-2908 Request permission

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
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 2873-2908
  • MSC (2020): Primary 53E20; Secondary 53C25, 58C05
  • DOI: https://doi.org/10.1090/tran/8295
  • MathSciNet review: 4223036