Paper · Aug 7, 2022

Moment map, convex function and extremal point

King Leung Lee, Jacob Sturm, Xiaowei Wang

math.DGarXiv:2208.03724
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Abstract

The moment map is a central concept in the study of Hamiltonian actions of compact Lie groups on symplectic manifolds. In this short note, we propose a theory of moment maps coupled with an -invariant convex function on , the dual of Lie algebra of , and study the properties of the critical point of . Our motivation comes from Donaldson which is an example of infinite dimensional version of our setting. As an application, we interpret Kähler-Ricci solitons as a special case of the generalized extremal metric.

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Submitted
Aug 7, 2022
Subjects
math.DG, math.AG, math.SG

Comments: 19 pages

Ricci solitonsKähler–Ricci solitonsKähler geometryHomogeneous spacesKähler–Einstein & K-stability

Its neighbourhood

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  1. Formal structure of scalar curvature in generalized Kähler geometry

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    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “moment map”, “Hamiltonian action”, “symplectic”

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    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “moment map”, “symplectic”, “Lie algebras”

  3. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

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  5. On the Existence of Weighted-cscK Metrics

    Jiyuan Han and Yaxiong Liu

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “extremal”

  6. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Hamiltonian action”, “critical points”

  7. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Homogeneous spaces

  8. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “extremal”

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) 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.