Paper · Nov 21, 2018

Structure of collapsing solutions of generalized Ricci flow

Steven Gindi, Jeffrey Streets

math.DGarXiv:1811.08877
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Abstract

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

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Submitted
Nov 21, 2018
Subjects
math.DG
Ricci flowGeneralized & G₂ flows(Non)collapsingStability & uniqueness

Its neighbourhood

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  1. Four-dimensional generalized Ricci flows with nilpotent symmetry

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  4. Optimal Transport and Generalized Ricci Flow

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  5. Generalized Ricci Flow

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  6. Bochner formulas, functional inequalities and generalized Ricci flow

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    Topics Generalized & G₂ flows · Same author Jeffrey Streets

  7. The canonical symmetry reduction of string backgrounds

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    Topics Generalized & G₂ flows, Stability & uniqueness · Same author Jeffrey Streets

  8. Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations

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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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.