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Classification results for expanding and shrinking gradient Kähler–Ricci solitons

Ronan J Conlon, Alix Deruelle and Song Sun

Geometry & Topology 28 (2024) 267–351
Abstract

We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler–Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler–Ricci solitons with quadratic curvature decay with derivatives. We then show that any two-dimensional complete shrinking gradient Kähler–Ricci soliton whose scalar curvature tends to zero at infinity is, up to pullback by an element of GL (2, ), either the flat Gaussian shrinking soliton on 2 or the U(2)–invariant shrinking gradient Kähler–Ricci soliton of Feldman, Ilmanen and Knopf on the blowup of 2 at one point. Finally, we show that up to pullback by an element of GL (n, ), the only complete shrinking gradient Kähler–Ricci soliton with bounded Ricci curvature on n is the flat Gaussian shrinking soliton, and on the total space of 𝒪(k) n1 for 0 < k < n is the U(n)–invariant example of Feldman, Ilmanen and Knopf. In the course of the proof, we establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler–Ricci soliton with bounded Ricci curvature in the Lie algebra of a torus. A key tool used to achieve this result is the Duistermaat–Heckman theorem from symplectic geometry. This provides the first step towards understanding the relationship between complete shrinking gradient Kähler–Ricci solitons and algebraic geometry.

Keywords
Kähler–Ricci solitons, Kähler–Ricci flow, self-similar solutions, Duistermaat–Heckman formula
Mathematical Subject Classification
Primary: 53C25
Secondary: 53C55, 53E30
References
Publication
Received: 25 March 2021
Revised: 27 February 2022
Accepted: 24 June 2022
Published: 27 February 2024
Proposed: Bruce Kleiner
Seconded: Tobias H Colding, Simon Donaldson
Authors
Ronan J Conlon
Department of Mathematical Sciences
The University of Texas at Dallas
Richardson, TX
United States
Alix Deruelle
Université Paris-Saclay
Laboratoire de mathématiques d’Orsay
Orsay
France
Song Sun
Institute for Advanced Study in Mathematics
Zhejiang University
Hangzhou
China
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States

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