Paper · May 28, 2024

Minimal hypersurfaces in by doubling the equatorial

Nikolaos Kapouleas, Jiahua Zou

math.DGWider flowsarXiv:2405.18283v2
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Abstract

For each large enough we construct by PDE gluing methods a closed embedded smooth minimal hypersurface doubling the equatorial three-sphere in , with containing bridges modelled after the three-dimensional catenoid and centered at the points of a square lattice contained in the Clifford torus . This answers a long-standing question of Yau in the case of and long-standing questions of Hsiang. Similarly we construct a self-shrinker of the Mean Curvature Flow in doubling the three-dimensional spherical self-shrinker with the bridges centered at the points of a square lattice contained in a Clifford torus . Both constructions respect the symmetries of the lattice as a subset of or and are based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of in . Furthermore converges as in the varifold sense to , and its volume .

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Submitted
May 28, 2024
Last revised
Aug 10, 2024 (v2)
Subjects
math.DG

Comments: 47 pages and no figures. This is an expanded version with further results on the volume and self-shrinkers

Mean curvature flowSelf-shrinkers & translators3-manifoldsWeak & singular flowsMinimal & CMC surfaces

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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.

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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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    Jingwen Chen and Pedro GasparWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  2. Self-expanders of positive genus

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  3. The global structure of bi-rotational self-shrinkers

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    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “Hsiang”, “Clifford”

  4. On the long-time limit of the mean curvature flow in closed manifolds

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    Topics Minimal & CMC surfaces, 3-manifolds

  5. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  6. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  7. The global structure of bi-rotational self-expanders

    Gregŕio Silva NetoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “Clifford”

  8. An eigenvalue estimate for self-shrinkers in a Ricci shirinker

    Franciele Conrado and Detang Zhou

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “minimal hypersurface”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

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

  2. 10A.2
    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.