On Perelman's Dilaton

TL;DR

This paper interprets Perelman's F-functional as an Einstein-Hilbert action in higher dimensions, linking Ricci flow to a constrained gravitational gradient flow through a Kaluza-Klein type approach.

Contribution

It introduces a novel higher-dimensional perspective on Perelman's F-functional, connecting Ricci flow to gravitational actions via a Kaluza-Klein framework.

Findings

Perelman's F-functional is equivalent to an Einstein-Hilbert action in extra dimensions.

Ricci flow corresponds to a constrained gradient flow of gravitational action.

Provides a geometric interpretation of Ricci flow in higher-dimensional gravity context.

Abstract

By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the ``real'' part of the metric under a constrained gradient flow of the Einstein-Hilbert gravitational action in higher dimension.