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Classification results
for expanding and shrinking gradient Kähler–Ricci
solitons
Ronan J Conlon, Alix Deruelle and Song Sun |
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Geometry & Topology 28 (2024) 267–351
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Abstract |
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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 , either the flat Gaussian shrinking soliton on or the –invariant shrinking gradient Kähler–Ricci soliton of Feldman, Ilmanen and Knopf on the blowup of at one point. Finally, we show that up to pullback by an element of , the only complete shrinking gradient Kähler–Ricci soliton with bounded Ricci curvature on is the flat Gaussian shrinking soliton, and on the total space of for is the –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
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Mathematical Subject Classification
Primary: 53C25
Secondary: 53C55, 53E30
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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
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Authors |
| Ronan J Conlon | |
| Department of Mathematical
Sciences The University of Texas at Dallas Richardson, TX United States |
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| Alix Deruelle | |
| Université Paris-Saclay Laboratoire de mathématiques d’Orsay Orsay France |
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| Song Sun | |
| Institute for Advanced Study in
Mathematics Zhejiang University Hangzhou China |
Department of Mathematics University of California, Berkeley Berkeley, CA United States |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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