Paper · Sep 28, 2026

Lagrangian mean curvature flow in the Kummer K3 surface

Carlos Alberto Ochoa Flores

math.DGWider flowsarXiv:2609.34923
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Abstract

A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.

Comments: 15 pages

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauLong-time existence & convergence

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    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Shared terms “special Lagrangian”

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    Topics Lagrangian mean curvature flow

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

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

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

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    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “Lagrangian spheres”, “special Lagrangian”

  5. Generic neck pinch singularities along 2D Lagrangian mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau · Shared terms “perturbation”

  6. Lagrangian mean curvature flow in the complex projective plane

    Christopher G. EvansWider flows PDF

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence

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

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau

  8. Mean Curvature Flows of Two-Convex Lagrangians

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    Topics Lagrangian mean curvature flow, Long-time existence & convergence

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.