Paper · Jul 19, 2017

Perelman's functionals on cones and Construction of type III Ricci flows coming out of cones

Tristan Ozuch

math.DGarXiv:1707.06102
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Abstract

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's and functionals of cones and characterize their finiteness in terms of the -functional of the link. As an application, we characterize manifolds with conical singularities on which a -functional can be defined and get upper bounds on the -functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theorem and prove that cones over some perturbations of the unit sphere can be smoothed out by type III immortal solutions on the Ricci flow.

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Submitted
Jul 19, 2017
Subjects
math.DG, math.AP

Comments: 54 pages

Ricci flowSingularitiesPerelman entropyWeak & singular flowsBoundaries & non-compactLong-time existence & convergence

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 Trace Anomaly, Perelman's Functionals and the Cosmological Constant (M. J. Luo, 2021)3Luo 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)5Chen 2022 The spinorial energy for asymptotically Euclidean Ricci flow (Julius Baldauf and Tristan Ozuch, 2022)6Baldauf 2022 Perelman's functionals on manifolds with non-isolated conical singularities (Xianzhe Dai and Changliang Wang, 2022)1Dai 2022 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)2Chow 2026 On the singularity formation of gauge fields coupled to Ricci flow (Andoni Royo Abrego, 2026)7Abrego 2026 Blowdowns along Ricci flow in dimension four (Dain Kim and Tristan Ozuch, 2026)8Kim 2026 On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Perelman's functionals on manifolds with non-isolated conical singularities

    Xianzhe Dai and Changliang Wang

    Topics Perelman entropy, Singularities · Shared terms “Perelman functionals”, “conical singularities”

  2. A Spinorial Perelman's Functional: Critical Points and Gradient Flow

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong

    Topics Perelman entropy · Shared terms “Perelman functionals”

  3. Trace Anomaly, Perelman's Functionals and the Cosmological Constant

    M. J. Luo

    Topics Perelman entropy · Shared terms “Perelman functionals”

  4. On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

    Yu Li, Longhuan Yan and Wenjia Zhang

    Topics Perelman entropy, Boundaries & non-compact, Singularities · Shared terms “asymptotically conical”

  5. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Weak & singular flows, Singularities · Shared terms “flow coming”

  6. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch

    Topics Boundaries & non-compact · Same author Tristan Ozuch · Shared terms “Perelman”

  7. On the singularity formation of gauge fields coupled to Ricci flow

    Andoni Royo Abrego

    Topics Perelman entropy, Singularities

  8. Blowdowns along Ricci flow in dimension four

    Dain Kim and Tristan Ozuch

    Topics Boundaries & non-compact, Singularities · Same author Tristan Ozuch · Shared terms “conical singularities”, “asymptotically conical”

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) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.