Paper · Jul 14, 2025

Short-time existence of Lagrangian mean curvature flow

Spandan Ghosh

math.DGarXiv:2507.10341v2
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Abstract

In his paper `Conjectures on Bridgeland Stability', Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez.

Comments: 61 pages, 3 figures. (v2) several typos corrected, added figures

Mean curvature flowLagrangian mean curvature flowRicci solitonsWeak & singular flowsStability & uniquenessHarmonic maps & DeTurck trickPhase fields & multiphase flows

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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 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)5Tsai 2023 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)1Bhattacharya 2024 Stability of the generalized Lagrangian mean curvature flow in cotangent bundle (Xishen Jin and Jiawei Liu, 2024)8Jin 2024 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)7Han 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)2Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)4Ghosh 2025 Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase (Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al., 2025)3Bhattacharya 2025 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)6Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

    Arunima Bhattacharya and Jeremy Wall PDF

    Topics Lagrangian mean curvature flow, Ricci solitons · Shared terms “expander”

  2. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Stability & uniqueness · Shared terms “Joyce”, “transverse”

  3. Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

    Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al.Wider flows PDF

    Topics Lagrangian mean curvature flow, Weak & singular flows

  4. Lagrangian mean curvature flow of surfaces with mean curvature bound

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow

  5. Entire solutions of two-convex Lagrangian mean curvature flows

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “time existence”, “stronger”

  6. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow

  7. Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

    Xiaoli Han, Jiayu Li and Jun SunWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “Joyce”

  8. Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

    Xishen Jin and Jiawei LiuWider flows PDF

    Topics Lagrangian mean curvature flow, Stability & uniqueness

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.