Paper · Jan 16, 2019

Ancient solutions in Lagrangian mean curvature flow

Ben Lambert, Jason D. Lotay, Felix Schulze

math.DGWider flowsarXiv:1901.05383v2
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As on arXiv

Abstract

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we classify Type II blow-ups of almost calibrated Lagrangian mean curvature flow when the blow-down is a pair of transverse planes or, when n=2, a multiplicity two plane. We also show that the Harvey-Lawson Clifford torus cone in C^3 cannot arise as the blow-down of an almost calibrated Type II blow-up.

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Submitted
Jan 16, 2019
Last revised
Apr 29, 2020 (v2)
Subjects
math.DG, math.AP, math.SG
Journal
Ann. Sc. Norm. Super. Pisa Cl. Sci. XXII (2021), 1169-1205

Comments: 30 pages, v2: minor typos corrected, accepted in Ann. Sc. Norm. Super. Pisa Cl. Sci

Mean curvature flowLagrangian mean curvature flowAncient solutionsSingularities

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 Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature (Keita Kunikawa, 2016)4Kunikawa 2016 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)3Evans 2018 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)7Lotay 2018 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)1Lotay 2022 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)2Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)8Lotay 2023 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)5Lotay 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 and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Ancient solutions · Same authors Jason D. Lotay, Felix Schulze · Shared terms “almost calibrated”

  2. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same authors Jason D. Lotay, Felix Schulze · Shared terms “transverse planes”, “Thomas Yau”

  3. Remarks on the self-shrinking Clifford torus

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

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

  4. Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

    Keita KunikawaWider flows

    Topics Lagrangian mean curvature flow, Ancient solutions · Shared terms “almost calibrated”, “calibrated Lagrangian”

  5. 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

  6. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

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

  7. Consequences of strong stability of minimal submanifolds

    Jason D. Lotay and Felix SchulzeWider flows

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

  8. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same author Jason D. Lotay · Shared terms “almost calibrated”, “calibrated Lagrangian”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.