The Long-Time Behavior of the Ricci Tensor Under The Ricci Flow

TL;DR

This paper investigates the long-term behavior of the Ricci tensor under Ricci flow on closed manifolds, showing it diminishes over time under certain conditions and classifying solutions in three dimensions.

Contribution

It establishes conditions under which the Ricci tensor vanishes at infinity and characterizes the type of solutions in three dimensions with bounded curvature and diameter.

Findings

Ricci tensor tends to zero as t approaches infinity for solutions with bounded curvature and diameter.

Solutions with bounded diameter and curvature product are of type III in three dimensions.

Provides conditions for the asymptotic behavior of Ricci flow solutions on closed manifolds.

Abstract

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as t goes to infinity. We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the square of the diameter with the norm of the curvature tensor is uniformly bounded, then the solution must be of type III.