Paper · Feb 24, 2020

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

Jason D. Lotay, Goncalo Oliveira

math.DGWider flowsarXiv:2002.10391v2
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Abstract

The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.

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Submitted
Feb 24, 2020
Last revised
Apr 3, 2022 (v2)
Subjects
math.DG, math.AP, math.SG

Comments: 45 pages, 9 figures; v2: minor changes, accepted version for JDG

Mean curvature flowCurve shorteningLagrangian mean curvature flowRicci-flat & Calabi–Yau4-manifoldsKähler geometryLong-time existence & convergenceMinimal & CMC surfaces

Its neighbourhood

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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 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 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)4Lambert 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)6Lotay 2020 The Gibbons-Hawking ansatz in generalized Kähler geometry (Jeffrey Streets and Yury Ustinovskiy, 2020)8Streets 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)3Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)1Lotay 2023 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)2Lotay 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)7Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Ricci-flat & Calabi–Yau, 4-manifolds · Same authors Jason D. Lotay, Goncalo Oliveira · Shared terms “special Lagrangian”, “circle invariant”

  2. 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 authors Jason D. Lotay, Goncalo Oliveira

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

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Same author Jason D. Lotay · Shared terms “Thomas Yau”

  4. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow · Same author Jason D. Lotay · Shared terms “Yau conjecture”, “Thomas Yau”

  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 author Jason D. Lotay · Shared terms “special Lagrangian”

  6. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Same author Jason D. Lotay

  7. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence · Shared terms “special Lagrangian”

  8. The Gibbons-Hawking ansatz in generalized Kähler geometry

    Jeffrey Streets and Yury Ustinovskiy

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “Hawking ansatz”, “Gibbons Hawking”

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.

Foundational paper in the Lagrangian mean curvature flow guide: Lotay–Oliveira: versions of the Thomas and Thomas–Yau conjectures for circle-invariant Lagrangians.

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.