Shrinking Gradient Kahler-Ricci Solitons and Toric Geometry
Channel: UMD Mathematics
Date: 03/03/2021

Speaker: Charles Cifarelli (UC Berkeley) -
Abstract: In this talk, I will present some recent work on the uniqueness of shrinking gradient K\"ahler-Ricci solitons on non-compact toric manifolds. In particular, the familiar Delzant classification holds in this context, and this allows one to apply the techniques of Berman-Berndtsson, Wang-Zhu, and others which reduce the problem to a study of a particular real Monge-Amp\`{e}re equation on a convex domain in \mathbb{R}^n. As a consequence, we will see that the standard product of the Fubini-Study metric on \mathbb{CP}^1 (round metric on S^2) and Euclidean metric on \mathbb{C} is the only shrinking gradient K\"ahler-Ricci soliton on \mathbb{CP}^1 \times \mathbb{C} with bounded scalar curvature.