Paper · Feb 5, 2018

Remarks on the self-shrinking Clifford torus

Christopher G. Evans, Jason D. Lotay, Felix Schulze

math.DGWider flowsarXiv:1802.01423v3
View PDF View HTML arXiv page

As on arXiv

Abstract

On the one hand, we prove that the Clifford torus in is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian -stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.

View PDF View HTML arXiv page
Submitted
Feb 5, 2018
Last revised
Jun 10, 2020 (v3)
Subjects
math.DG, math.AP, math.SG
Journal
J. Reine Angew. Math. 2020 (2020), 139-170
DOI
10.1515/crelle-2019-0015

Comments: 31 pages, v3: additional details for proof of local uniqueness of the Clifford torus as a self-shrinker provided

Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsStability & uniqueness

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 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)4Lotay 2018 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)1Lambert 2019 On the entire self-shrinking solutions to Lagrangian mean curvature flow II (Rongli Huang, Qianzhong Ou and Wenlong Wang, 2019)7Huang 2019 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)3Evans 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)6Lotay 2020 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)2Lotay 2022 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)5Lotay 2022 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)8Lotay 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow · Same authors Jason D. Lotay, Felix Schulze · Shared terms “Clifford torus”

  2. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same authors Jason D. Lotay, Felix Schulze

  3. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Christopher G. Evans · Shared terms “shrinking Clifford”, “Clifford torus”, “self shrinking”

  4. Consequences of strong stability of minimal submanifolds

    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. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

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

  6. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow · Same author Jason D. Lotay · Shared terms “Clifford torus”

  7. On the entire self-shrinking solutions to Lagrangian mean curvature flow II

    Rongli Huang, Qianzhong Ou and Wenlong WangWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “self shrinking”, “proof relies”

  8. 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 · 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) 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) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.