Paper · Mar 30, 2020

On instability of Type (II) Lawson-Osserman Cones

Zhaohu Nie, Yongsheng Zhang

math.DGWider flowsarXiv:2003.13286
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Abstract

We obtain the instability of Type (II) Lawson-Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) unstable solutions with singularity to the Dirichlet problem for minimal graphs of high codimension versus smooth unstable ones by Lawson-Osserman through a min-max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs and unlikely detectible through the mean curvature flow or min-max theory.

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Submitted
Mar 30, 2020
Subjects
math.DG

Comments: 15 pages, 6 figures

Mean curvature flowSingularitiesWeak & singular flowsStability & uniquenessMinimal & CMC surfaces

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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
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    Chen-Kuan LeeWider flows

    Topics Mean curvature flow · Shared terms “Lawson Osserman”, “Osserman cone”

  2. Existence and non-existence of minimal graphs

    Qi Ding, J. Jost and Y. L. XinWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “minimal graphs”, “Dirichlet”

  3. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows

    Topics Minimal & CMC surfaces, Weak & singular flows, Singularities

  4. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Topics Minimal & CMC surfaces · Shared terms “min max”, “max theory”

  5. Dirichlet problem for -minimal graphs

    Jean-Baptiste Casteras, Esko Heinonen and Ilkka HolopainenWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal graphs”, “Dirichlet”

  6. Multiplicity one for min-max theory in compact manifolds with boundary and its applications

    Ao Sun, Zhichao Wang and Xin ZhouWider flows

    Topics Minimal & CMC surfaces · Shared terms “min max”, “max theory”, “Euclidean space”

  7. Minimal two-spheres in three-spheres

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces · Shared terms “min max”, “max theory”

  8. Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

    Katherine Castro and César RosalesWider flows

    Topics Minimal & CMC surfaces, Singularities, Stability & uniqueness · Shared terms “Euclidean space”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.