Dimensional reduction and the long-time behavior of Ricci flow

TL;DR

This paper studies the long-term behavior of three-dimensional Ricci flows with specific curvature decay and diameter growth conditions, showing they tend toward a homogeneous expanding soliton on the universal cover.

Contribution

It establishes conditions under which Ricci flow solutions converge to homogeneous expanding solitons, advancing understanding of their asymptotic geometric behavior.

Findings

Ricci flow solutions approach homogeneous expanding solitons

Curvature decays like 1/t, diameter grows like sqrt(t)

Universal cover analysis reveals asymptotic homogeneity

Abstract

If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that decay like the inverse of t and diameter that increases at most like the square root of t, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.