On the asymptotic reduced volume of the Ricci flow

TL;DR

This paper demonstrates that two different monotone quantities related to the Ricci flow have the same asymptotic limit for ancient solutions, linking Perelman's reduced volume with a local quantity by Ecker, Knopf, Ni, and Topping.

Contribution

It establishes the equality of asymptotic limits of two monotone quantities in Ricci flow, connecting Perelman's reduced volume with a local geometric quantity.

Findings

Asymptotic limits of the two quantities coincide for ancient solutions.

The relation between Perelman's reduced volume and the local quantity is established.

Provides new insights into the behavior of Ricci flow solutions over time.

Abstract

In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This establishes a relation between these two monotone quantities.