Paper · May 30, 2016

Concavity of Perelman's -functional over the space of Kähler potentials

Nefton Pali

math.DGarXiv:1605.09290
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Abstract

In this short note we observe that the concavity of Perelman's -functional over a neighborhood of a Kähler-Ricci soliton inside the space of Kähler potentials is a direct consequence of author's solution of the variational stability problem for Kähler-Ricci solitons. Independently, we provide a rather simple proof of this fact based on some elementary formulas obtained in our previous work.

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Submitted
May 30, 2016
Subjects
math.DG
Journal
European Journal of Mathematics, 3 (2017), no. 3, 587-602
Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness

Its neighbourhood

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    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “variational”

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  4. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

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  5. Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature

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    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “short note”

  6. Linear Stability of Steady and Expanding Kähler-Ricci Solitons

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

  7. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

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

  8. Kähler-Ricci solitons on horospherical variety

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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) 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.