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arXiv · 2501.12949

Deriving Perelman's entropy from Colding's monotonic volume

Ignacio Bustamante·Martin Reiris
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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 $N\rightarrow \infty$, 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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Ignacio Bustamante, Martin Reiris. 2025-01-22. Deriving Perelman's entropy from Colding's monotonic volume. https://doi.org/10.1515/crelle-2025-0034

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