Paper · Sep 21, 2021

Variational structure and uniqueness of generalized Kähler-Ricci solitons

Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:2109.10295
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Abstract

Under broad hypotheses we derive a scalar reduction of the generalized Kähler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the classification in complex dimension .

Comments: 35 pages, comments are welcome

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness

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    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  2. Formal structure of scalar curvature in generalized Kähler geometry

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    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Hamiltonian action”

  3. The generalized Kähler Calabi-Yau problem

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    Topics Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Hamiltonian”

  4. On geometry of steady toric Kähler-Ricci solitons

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    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Yury Ustinovskiy

  5. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

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    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness

  7. From Kähler Ricci solitons to Calabi-Yau Kähler cones

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  8. Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

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    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness

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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  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.