Deriving Perelman's entropy from Colding's monotonic volume

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 $N\to \mathrm{\infty }$, 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.