Paper · Dec 17, 2021

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

James Stanfield

math.DGarXiv:2112.09344
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Abstract

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

Comments: 13 pages

Generalized & G₂ flowsHomogeneous spacesStability & uniqueness

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    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Positive Hermitian”, “Hermitian curvature”, “Lie groups”

  2. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

    Topics Generalized & G₂ flows, Homogeneous spaces, Stability & uniqueness · Same author James Stanfield · Shared terms “Lie groups”, “Killing form”, “dynamically unstable”

  3. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author James Stanfield · Shared terms “Lie groups”

  4. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “geometric flows”

  5. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”, “Lie groups”

  6. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “algebraic”, “Lie”

  7. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “dynamically”

  8. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

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    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Killing form”, “algebraic”, “Lie”

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Background reading

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

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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