Paper · Feb 12, 2018

Consequences of strong stability of minimal submanifolds

Jason D. Lotay, Felix Schulze

math.DGWider flowsarXiv:1802.03941v2
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Abstract

In this note we show that the recent dynamical stability result for small -perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow which are proved to occur by Neves.

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Submitted
Feb 12, 2018
Last revised
Apr 19, 2018 (v2)
Subjects
math.DG, math.AP
Journal
IMRN 2020 (2020), 2352-2360
DOI
10.1093/imrn/rny095

Comments: 7 pages, minor corrections. Final version to appear in IMRN

Mean curvature flowLagrangian mean curvature flowSingularitiesWeak & singular flowsStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

Connected papers

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 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A strong stability condition on minimal submanifolds and its implications (Chung-Jun Tsai and Mu-Tao Wang, 2017)3Tsai 2017 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)4Evans 2018 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)2Lambert 2019 Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz (Jason D. Lotay and Goncalo Oliveira, 2020)8Lotay 2020 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)7Lotay 2020 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)5Lotay 2022 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)1Lotay 2022 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)6Lotay 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Weak & singular flows, Singularities · Same authors Jason D. Lotay, Felix Schulze

  2. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same authors Jason D. Lotay, Felix Schulze

  3. A strong stability condition on minimal submanifolds and its implications

    Chung-Jun Tsai and Mu-Tao WangWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “Strong Stability”, “minimal submanifolds”, “dynamical stability”

  4. Remarks on the self-shrinking Clifford torus

    Christopher G. Evans, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Stability & uniqueness · Same authors Jason D. Lotay, Felix Schulze · Shared terms “perturbation”

  5. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow · Same authors Jason D. Lotay, Felix Schulze

  6. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same author Jason D. Lotay

  7. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same author Jason D. Lotay

  8. Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Minimal & CMC surfaces · Same author Jason D. Lotay

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) 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) 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)

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.