Paper · Dec 26, 2023

The Smale Conjecture and Minimal Legendrian Graph in

Shu-Cheng Chang, Chin-Tung Wu, Liuyang Zhang

math.DGWider flowsarXiv:2312.15996v3
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Abstract

In this article, we recapture the Smale conjecture on a Sasakian -sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian -spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in . By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in . Finally, we will address the rigidity theorem of -dimensional Legendrian self-shrinkers in . We are able to reconstruct the Harvey-Lawson special Lagrangian cone in from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.

Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesSmale conjecture & diffeomorphismsStability & uniquenessSasakian & contact geometry

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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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  1. Legendrian mean curvature flow in -Einstein Sasakian manifolds

    Shu-Cheng Chang, Yingbo Han and Chin-Tung WuWider flows PDF

    Topics Lagrangian mean curvature flow, Sasakian & contact geometry, Singularities · Same authors Shu-Cheng Chang, Chin-Tung Wu · Shared terms “Legendrian”, “special Lagrangian”, “Lagrangian cones”

  2. Rigidity theorems for complete Legendrian self-shrinkers

    Shu-Cheng Chang, Hongbing Qiu and Liuyang ZhangWider flows PDF

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Stability & uniqueness · Same authors Shu-Cheng Chang, Liuyang Zhang · Shared terms “Legendrian self”, “Legendrian mean”, “rigidity theorem”

  3. Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

    Shu-Cheng Chang, Chien Lin and Chin-Tung Wu PDF

    Topics Sasakian & contact geometry, Singularities · Same authors Shu-Cheng Chang, Chin-Tung Wu

  4. On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian -manifold

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Singularities · Same authors Shu-Cheng Chang, Chin-Tung Wu

  5. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Singularities · Same authors Shu-Cheng Chang, Chin-Tung Wu

  6. Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

    Der-Chen Chang, Shu-Cheng Chang, Yingbo Han et al. PDF

    Topics Sasakian & contact geometry, Singularities · Same authors Shu-Cheng Chang, Chin-Tung Wu

  7. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al. PDF

    Topics Sasakian & contact geometry · Same authors Shu-Cheng Chang, Chin-Tung Wu

  8. The Smale conjecture and min-max theory

    Daniel Ketover and Yevgeny Liokumovich PDF

    Topics Smale conjecture & diffeomorphisms

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) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) 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) 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)

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.