Paper · Jan 22, 2018

The branching curves and their application to the four dimensional Ricci flow

Ilias Tergiakidis

math.DGarXiv:1801.07315v2
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Abstract

We study the four dimensional Ricci flow with the help of local invariants. If is a solution to the Ricci flow and , we can associate to the point a one-parameter family of curves, which lie in the product of two projective lines. This allows us to reformulate the Cheeger-Gromov-Hamilton Compactness Theorem in the context of these curves. We use this result, in order to study Type I singularities in dimension four and give a characterization of the corresponding singularity models.

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Submitted
Jan 22, 2018
Last revised
Aug 23, 2018 (v2)
Subjects
math.DG

Comments: Corrected typos

Ricci flowSingularities4-manifoldsLimit spaces & RCD

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  4. On an invariant curvature cone along 4-dimensional Ricci flow

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  5. Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

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    Topics 4-manifolds · Shared terms “four dimensional”

  6. Singular sets in noncollapsed Ricci flow limit spaces

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    Topics 4-manifolds, Limit spaces & RCD, Singularities · Shared terms “four dimensional”, “dimension four”, “Gromov”

  7. On the classification of four-dimensional gradient Ricci solitons

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    Topics 4-manifolds · Shared terms “four dimensional”

  8. Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

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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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.