Convergence of Ricci flow solutions to Taub-NUT

TL;DR

This paper investigates the long-term behavior of Ricci flow solutions on certain 4-manifolds, proving convergence to the Taub-NUT metric under specific initial conditions and classifying asymptotically flat cases.

Contribution

It establishes convergence criteria for Ricci flow to the Taub-NUT metric and provides a uniqueness result for the Taub-NUT solution as a singularity model.

Findings

Ricci flow converges to Taub-NUT under specified conditions

Classification of long-time behavior for asymptotically flat initial metrics

Uniqueness of the Taub-NUT metric as an infinite-time singularity model

Abstract

We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$ on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If $g_{0}$ has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric $g_{\mathsf{TNUT}}$ in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when $g_{0}$ is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for $g_{\mathsf{TNUT}}$.