Paper · Oct 16, 2020

Cohomogeneity one Kähler-Ricci solitons under a Heisenberg group action and related metrics

Gideon Maschler, Robert Ream

math.DGarXiv:2010.09218v4
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Abstract

We show that integrability of an almost complex structure in complex dimension is equivalent, in the presence of an almost hermitian metric, to equations involving what we call shear operators. Inspired by this, we give an ansatz for Kähler metrics in dimension , for which at most of these shear equations are non-trivial. The equations for gradient Kähler-Ricci solitons in this ansatz are frame dependent PDEs, which specialize to ODEs under extra assumptions. Metrics solving the latter system include a restricted class of cohomogeneity one metrics, and we find among them complete expanding gradient Kähler-Ricci solitons under the action of the -dimensional Heisenberg group, and some incomplete steady solitons. We examine curvature properties and asymptotics for the former Ricci solitons. In another special case of the ansatz we present, for , a class of complete metrics of a more general type which we call gradient Kähler-Ricci skew-solitons, which are cohomogeneity one under the Euclidean plane group action. This paper continues research started in [MR, AM2].

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Submitted
Oct 16, 2020
Last revised
Jan 5, 2021 (v4)
Subjects
math.DG

Comments: In this version we correct a few statements related to curvature and distance, and add material on asymptotics (section 5.5) and on curvature bounds. arXiv admin note: text overlap with arXiv:2007.06471

Ricci solitonsKähler–Ricci solitonsKähler geometryHomogeneous spacesAnalytic minimal model program

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 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.