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Deriving Perelman’s entropy from Colding’s monotonic volume

  • Ignacio Bustamante ORCID logo EMAIL logo und Martín Reiris
Veröffentlicht/Copyright: 29. Mai 2025

Abstract

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop–Gromov volume comparison applied to a suitably constructed 𝑁-space, which becomes Ricci-flat as N , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman’s entropy emerges as the limit of Colding’s monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman’s 𝑁-space.

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Received: 2025-04-14
Published Online: 2025-05-29
Published in Print: 2025-09-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 22.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/crelle-2025-0034/html
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