Paper · Jan 20, 2016

New type I ancient compact solutions of the Yamabe flow

Panagiota Daskalopoulos, Manuel del Pino, John King, Natasa Sesum

math.DGWider flowsarXiv:1601.05349
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As on arXiv

Abstract

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as , to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

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Submitted
Jan 20, 2016
Subjects
math.DG

Comments: arXiv admin note: substantial text overlap with arXiv:1509.08803

Yamabe flowAncient solutionsSingularitiesBoundaries & non-compactRotational symmetry & warped products

Its neighbourhood

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 2016, 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 Existence and properties of ancient solutions of the Yamabe flow (Shu-Yu Hsu, 2016)6Hsu 2016 Yamabe Flow: steady solitons and type II singularities (Beomjun Choi and Panagiota Daskalopoulos, 2017)4Choi 2017 Uniqueness of two-convex closed ancient solutions to the mean curvature flow (Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2018)3Angenent 2018 Type II Singularities on complete non-compact Yamabe flow (Beomjun Choi, Panagiota Daskalopoulos and John King, 2018)1Choi 2018 Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2019)5Angenent 2019 Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Panagiota Daskalopoulos and Natasa Sesum, 2019)2Daskalopoulo 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)8Brendle 2020 Dynamics of Convex Mean Curvature Flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2023)7Angenent 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type II Singularities on complete non-compact Yamabe flow

    Beomjun Choi, Panagiota Daskalopoulos and John King

    Topics Yamabe flow, Rotational symmetry & warped products, Boundaries & non-compact · Same authors Panagiota Daskalopoulos, John King · Shared terms “rotationally symmetric”

  2. Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Panagiota Daskalopoulos and Natasa Sesum

    Topics Ancient solutions, Singularities · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “ancient compact”

  3. Uniqueness of two-convex closed ancient solutions to the mean curvature flow

    Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Rotational symmetry & warped products, Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “rotationally symmetric”

  4. Yamabe Flow: steady solitons and type II singularities

    Beomjun Choi and Panagiota Daskalopoulos

    Topics Yamabe flow, Boundaries & non-compact, Singularities · Same author Panagiota Daskalopoulos

  5. Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “ancient compact”

  6. Existence and properties of ancient solutions of the Yamabe flow

    Shu-Yu HsuWider flows

    Topics Yamabe flow, Ancient solutions

  7. Dynamics of Convex Mean Curvature Flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Boundaries & non-compact · Same authors Panagiota Daskalopoulos, Natasa Sesum

  8. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics Ancient solutions, Singularities · Same authors Panagiota Daskalopoulos, Natasa Sesum

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.