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Brett Kotschwar, Arizona State University – Analysis & PDE Seminar
March 6, 2019 @ 4:00 pm - 5:00 pm
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.