Paper · Aug 22, 2017

Minimal two-spheres in three-spheres

Robert Haslhofer, Daniel Ketover

math.DGWider flowsarXiv:1708.06567v2
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Abstract

We prove that any manifold diffeomorphic to and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in . Finally, considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

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Submitted
Aug 22, 2017
Last revised
Sep 6, 2017 (v2)
Subjects
math.DG, math.AP, math.GT
Journal
Duke Math. J. 168, no. 10 (2019), 1929-1975
DOI
10.1215/00127094-2019-0009

Comments: 41 pages

Mean curvature flow3-manifoldsMinimal & CMC surfaces

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
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    Topics Minimal & CMC surfaces · Same authors Robert Haslhofer, Daniel Ketover · Shared terms “combines ideas”, “convex foliations”, “generic metric”

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

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    Topics Minimal & CMC surfaces · Same author Daniel Ketover · Shared terms “max theory”, “min max”

  4. Rigidity of riemannian manifolds containing an equator

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    Topics Minimal & CMC surfaces · Shared terms “Simon Smith”, “minimal spheres”

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

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    Topics Minimal & CMC surfaces, 3-manifolds · Shared terms “embedded minimal”, “multiplicity”

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

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

  7. Free boundary flow with surgery

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    Topics Minimal & CMC surfaces · Same author Robert Haslhofer · Shared terms “mean convex”

  8. Minimal hypersurfaces in by doubling the equatorial

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    Topics Minimal & CMC surfaces, 3-manifolds · Shared terms “centered”, “Yau”

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.