Paper · Jan 26, 2022

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

Letian Chen

math.DGWider flowsarXiv:2201.11179v2
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Abstract

We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity.

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Submitted
Jan 26, 2022
Last revised
Dec 25, 2022 (v2)
Subjects
math.DG

Comments: 32 pages; Fixed a gap in the proof of Proposition 6.1. To appear in Calc. Var. Partial Differential Equations

Mean curvature flowSingularitiesWeak & singular flowsRotational symmetry & warped products

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows

    Letian ChenWider flows

    Topics Weak & singular flows · Same author Letian Chen · Shared terms “flow coming”

  2. Rotationally symmetric Ricci Flow on

    Ming Hsiao

    Topics Rotational symmetry & warped products, Weak & singular flows, Singularities · Shared terms “rotationally symmetric”, “short time”

  3. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows · Shared terms “rotationally symmetric”

  4. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”

  5. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “time singularity”, “finite time”

  6. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

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

  7. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  8. Singular Rotational Self-Similar Tori for Odd -Curvature Flows

    Haoxuan Cheng, Junqi Lai and Guoxin WeiWider flows

    Topics Rotational symmetry & warped products

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.