Thesis Defense - Michael Law

Speaker: Michael Law Title: Symmetry, Stability, and Classification of Ricci Solitons Abstract: Ricci solitons provide models for singularity formation and resolution in Ricci flow, while Ricci-flat asymptotically locally Euclidean (ALE) spaces arise as local models for the degeneration and desingularization of Einstein metrics. In this thesis, we develop a symmetry principle for steady and expanding gradient Ricci solitons and Ricci-flat ALE spaces under suitable asymptotic conditions. The principle relates symmetries at infinity to global isometries through dimension bounds on the kernel of the relevant Lichnerowicz operator. This kernel measures an obstruction to extending asymptotic symmetries and is closely connected to linear stability. We apply this principle to characterize the Bryant solitons and the four-dimensional Appleton solitons under analytical hypotheses implied by strict stability. For the Appleton characterization, we also establish uniqueness within Appleton’s defining ansatz and strict stability of the four-dimensional examples. More generally, we classify four-dimensional steady gradient Ricci solitons with vanishing weighted Lichnerowicz kernel that are asymptotically cylindrical with link SU(2)/G, where G is a finite subgroup of SU(2), showing that they are quotients of Bryant or Appleton solitons. In the Ricci-flat ALE setting, bounds on the Lichnerowicz kernel yield characterizations of Eguchi–Hanson space and its higher-dimensional Calabi analogs. Zoom Link: https://mit.zoom.us/j/92057992284