Paper · Jul 16, 2023

Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

Albert Chau, Adam Martens

math.DGarXiv:2307.08088v2
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Abstract

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow emerging from an arbitrary 3D complete noncompact Riemannian manifold which has nonnegative Ricci curvature. We show is complete for positive times provided satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show is complete for positive times provided is a compactly supported perturbation of a nonnegative sectional curvature metric on .

Comments: 12 pages; statement of Theorem 1.2 revised; correction and details added in proof of Theorem 1.2 (section 4); references [8], [18], [24] added

Ricci flowOther curvature flows3-manifoldsWeak & singular flowsBoundaries & non-compact

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)3Kleiner 2014 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)7Deruelle 2022 Inverse mean curvature flow and Ricci-pinched three-manifolds (Gerhard Huisken and Thomas Koerber, 2023)8Huisken 2023 Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature (Albert Chau and Adam Martens, 2024)2Chau 2024 Ends of (singular) Ricci shrinkers (Alessandro Bertellotti and Reto Buzano, 2024)6Bertellotti 2024 Sharpening a gap theorem: nonnegative Ricci and small curvature concentration (Adam Martens, 2024)4Martens 2024 A note on Ricci flow from small curvature concentration and a Morrey-type condition (Albert Chau and Adam Martens, 2026)1Chau 2026 On the uniqueness of Ricci flows from certain Alexandrov spaces (Laura Flower, 2026)5Flower 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on Ricci flow from small curvature concentration and a Morrey-type condition

    Albert Chau and Adam Martens PDF

    Topics Boundaries & non-compact · Same authors Albert Chau, Adam Martens · Shared terms “Riemannian manifolds”

  2. Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

    Albert Chau and Adam Martens PDF

    Topics Boundaries & non-compact · Same authors Albert Chau, Adam Martens

  3. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics 3-manifolds, Weak & singular flows

  4. Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

    Adam Martens PDF

    Topics Ricci flow · Same author Adam Martens · Shared terms “nonnegative Ricci”, “volume ratio”

  5. On the uniqueness of Ricci flows from certain Alexandrov spaces

    Laura Flower PDF

    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “Lai”, “sectional curvature”, “complete noncompact”

  6. Ends of (singular) Ricci shrinkers

    Alessandro Bertellotti and Reto Buzano PDF

    Topics Weak & singular flows, Boundaries & non-compact

  7. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics 3-manifolds, Weak & singular flows · Shared terms “Lott”, “Riemannian manifolds”

  8. Inverse mean curvature flow and Ricci-pinched three-manifolds

    Gerhard Huisken and Thomas Koerber PDF

    Topics 3-manifolds, Boundaries & non-compact, Other curvature flows · Shared terms “Topping”, “Lott”, “Simon”

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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.

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

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    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

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    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.