Asymptotically cylindrical steady Kähler-Ricci solitons

Johannes Schäfer

Paper · Mar 23, 2021

Asymptotically cylindrical steady Kähler-Ricci solitons

Johannes Schäfer

math.DGarXiv:2103.12629v2

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Abstract

Let $D$ be a compact Kähler manifold with trivial canonical bundle and $Γ$ be a finite cyclical group of order $m$ acting on $C \times D$ by biholomorphisms, where the action on the first factor is generated by rotation of angle $2 π / m$ . Furthermore, suppose that $Ω_{D}$ is a trivialisation of the canonical bundle such that $Γ$ preserves the holomorphic form $d z \land Ω_{D}$ on $C \times D$ , with $z$ denoting the coordinate on $C$ . The main result of this article is the construction of new examples of gradient steady Kähler-Ricci solitons on certain crepant resolutions of the orbifolds $(C \times D) /Γ$ . These new solitons converge exponentially to a Ricci-flat cylinder $R \times (S^{1} \times D) /Γ$ .

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Submitted: Mar 23, 2021
Last revised: Aug 15, 2021 (v2)
Subjects: math.DG

Comments: 64 pages; results unchanged, but details added to computations in Section 5

Ricci solitons Kähler–Ricci solitons Ricci-flat & Calabi–Yau Kähler geometry Long-time existence & convergence

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[1] 11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

[2] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

Asymptotically cylindrical steady Kähler-Ricci solitons

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