Communications in Analysis and Geometry

Volume 17 (2009)

Number 2

Perelman’s reduced volume and a gap theorem for the Ricci flow

Pages: 227 – 263



Takumi Yokota (Graduate School of Pure and Applied Sciences, University of Tsukuba, Japan)


In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson’s result for Ricci-flat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained.

Published 1 January 2009