Paper · May 25, 2023

Hamiltonian $2$ -forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on $C^{n}$

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Abstract

For each partition of the positive integer $n = ℓ + \sum_{j = 1}^{ℓ} d_{j}$ , where $ℓ \geq 1$ and $d_{j} \geq 0$ are integers, we construct a continuous $(ℓ - 1)$ -parameter family of explicit complete gradient steady Kähler–Ricci solitons on $C^{n}$ admitting a hamiltonian $2$ -form of order $ℓ$ and symmetry group $U (d_{1} + 1) \times \dots \times U (d_{ℓ} + 1)$ . For $ℓ = 1, d_{1} = n - 1$ we obtain Cao's example [17] whereas for other partitions the metrics are new. Furthermore, when $n = 2, ℓ = 2, d_{1} = d_{2} = 0$ we obtain complete gradient steady Kähler–Ricci solitons on $C^{2}$ which have positive sectional curvature but are not isometric to Cao's $U (2)$ -invariant example. This disproves a conjecture by Cao. We also present a construction yielding explicit families of complete gradient steady Kähler-Ricci solitons on $C^{n}$ containing higher dimensional extensions of the Taub-NUT Ricci-flat Kähler metric on $C^{2}$ . When $n \geq 3$ , the complete Ricci-flat Kähler metrics, and when $n \geq 2$ , their deformations to complete gradient steady Kähler Ricci solitons seem not to have been observed before our work.

Comments: final version, to appear in JDG

Ricci solitons Kähler–Ricci solitons Ricci-flat & Calabi–Yau Sphere theorems Kähler geometry

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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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
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.

Hamiltonian $2$ -forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on $C^{n}$

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Hamiltonian 2-forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on Cn

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Hamiltonian $2$ -forms and new explicit Calabi–Yau metrics and gradient steady Kähler–Ricci solitons on $C^{n}$