Ricci Flow on ALF manifolds
김수현
129동 301호
0
1965
08.10 17:04
| 구분 | DASOM |
|---|---|
| 일정 | 2026-09-08(화) 13:30~15:00 |
| 세미나실 | 129동 301호 |
| 강연자 | 김다인 (MIT) |
| 담당교수 | 임선희 |
| 기타 | DASOM 교류세미나 |
화요일 오후 1시반-3시 (1:30-2:30 강연 + 2:30-2:50 Q&A)
제목: Ricci Flow on ALF manifolds
초록: 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 is joint work with Tristan Ozuch.
