The canonical shrinking soliton associated to a Ricci flow

TL;DR

This paper constructs a canonical shrinking Ricci soliton from a given Ricci flow, linking properties of the original flow to a higher-dimensional soliton via optimal transportation theory.

Contribution

It introduces a new geometric construction associating a shrinking Ricci soliton to any Ricci flow, expanding the understanding of their relationship through optimal transportation.

Findings

Establishes a correspondence between Ricci flow and higher-dimensional Ricci soliton.

Connects Ricci flow properties to optimal transportation theory.

Provides a framework for analyzing Ricci flows via associated solitons.

Abstract

To every Ricci flow on a manifold M over a time interval I, we associate a shrinking Ricci soliton on the space-time M x I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author, and McCann and the second author; we briefly survey the link between these subjects.