Paper · Mar 6, 2024

A family of Kähler flying wing steady Ricci solitons

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Abstract

In $1996$ , H.-D. Cao constructed a $U (n)$ -invariant steady gradient Kähler-Ricci soliton on $C^{n}$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on $C^{n}$ is necessarily $U (n)$ -invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n = 2$ . Here, we construct a family of $U (1) \times U (n - 1)$ -invariant, but not $U (n)$ -invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real $(1, 1)$ -forms (in particular, with strictly positive sectional curvature) on $C^{n}$ for $n \geq 3$ , thereby answering Cao's question in the negative for $n \geq 3$ . This family of steady Ricci solitons interpolates between Cao's $U (n)$ -invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's $U (n - 1)$ -invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of $P^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1, 1)$ -forms.

Comments: 53 pages, various corrections, final version

Ricci solitons Kähler–Ricci solitons Ancient solutions Sphere theorems Kähler geometry Stability & uniqueness

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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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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

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.
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.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.

A family of Kähler flying wing steady Ricci solitons

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