Brett Kotschwar, Arizona State University – Analysis & PDE Seminar

Title: The determination of a shrinking Ricci soliton from its geometry at infinity.

Abstract: Shrinking solitons are self-similar solutions to the Ricci flow and models for the geometry of a solution near a developing singularity. The geometric behavior of a complete noncompact shrinking soliton near infinity is highly constrained; in dimension four, it has been conjectured that every such soliton is either smoothly asymptotic to a cone or to a (quotient of) a generalized cylinder.  I will describe some uniqueness results, obtained jointly with Lu Wang,  which demonstrate the extent to which a shrinking soliton is determined by its asymptotic geometry, and discuss their application to the four-dimensional classification.  These results are obtained by transforming the problem — on its face, an elliptic problem of unique continuation at infinity — into a finite-time problem of unique continuation for a parabolic system.