Deriving Perelman's entropy from Colding's monotonic volume

TL;DR

This paper shows that Perelman's entropy can be derived as a limit of Colding's monotonic volume in Ricci-flat manifolds, linking two fundamental monotonic quantities in geometric analysis.

Contribution

It provides a novel explanation for the origin of Perelman's entropy by connecting it to Colding's monotonic volume in the setting of Ricci-flat manifolds.

Findings

Perelman's entropy arises as a limit of Colding's volume

The connection is established in the context of Ricci-flat manifolds

The result clarifies the geometric origin of Perelman's entropy

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.