Dain Kim (MIT) will give a talk on 03/31/2026.

Following the seminal work of Hamilton and Perelman, Ricci flow has proved to be a powerful tool for understanding the topology and geometry of 3D manifolds. A natural next question is whether Ricci flow can play a similar role in 4D. We study the long-time behavior of Ricci flow on 4D manifolds, focusing on Ricci-flat asymptotically locally flat (ALF) spaces. These manifolds form the simplest collapsing Ricci-flat 4-manifolds and are widely conjectured to arise as singularity models for the long-time behavior of Ricci flow.

We develop a framework for Ricci flow on ALF manifolds. First, we show that the ALF structure is preserved along the flow. We then introduce a renormalized version of Perelman’s $\lambda$-functional adapted to the ALF setting, defined using a notion of relative mass with respect to a fixed Ricci-flat reference metric. Within this framework, we show that Ricci flow is the gradient flow of this adapted $\lambda$-functional in a weighted $L^2$ sense. This variational structure allows us to define and analyze linear and dynamical stability for Ricci-flat ALF metrics.

As an application, we show that conformally Kähler but non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. This instability is significant for long-time analysis, as it suggests that such metrics are dynamically disfavored as singularity models. A key ingredient is a Fredholm theory for the Laplacian on weighted Hölder spaces over ALF manifolds.

This talk assumes only a basic background in differential geometry and is intended to be accessible to a broad graduate audience.