Paper · Mar 12, 2024

Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

Arunima Bhattacharya, Jeremy Wall

math.AParXiv:2403.07235
PDF Abstract page

As on arXiv

Abstract

In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.

Comments: 28 pages

Mean curvature flowLagrangian mean curvature flowRicci solitonsSelf-shrinkers & translators

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 2021, 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 and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)4Lotay 2022 Cohomogeneity-One Lagrangian Mean Curvature Flow (Jesse Madnick and Albert Wood, 2022)7Madnick 2022 Rotators-Translators to Mean Curvature Flow in ℍ²×ℝ (Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos Santos, 2024)6de Lima 2024 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)2Bhattacharya 2024 A Liouville type theorem for ancient Lagrangian mean curvature flows (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2024)3Bhattacharya 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)5Li 2024 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)8Han 2024 Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase (Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al., 2025)1Bhattacharya 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Arunima Bhattacharya, Jeremy Wall · Shared terms “Lagrangian phase”, “type equations”, “curvature type”

  2. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows PDF

    Topics Lagrangian mean curvature flow · Same authors Arunima Bhattacharya, Jeremy Wall · Shared terms “Lagrangian phase”, “type equations”, “curvature type”

  3. A Liouville type theorem for ancient Lagrangian mean curvature flows

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Arunima Bhattacharya

  4. Ancient solutions and translators of Lagrangian mean curvature flow

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

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators

  5. Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

    Shanshan Li, Jiaru Lv and Rongli HuangWider flows PDF

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “type equations”

  6. Rotators-Translators to Mean Curvature Flow in

    Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos SantosWider flows PDF

    Topics Self-shrinkers & translators

  7. Cohomogeneity-One Lagrangian Mean Curvature Flow

    Jesse Madnick and Albert Wood PDF

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Ricci solitons

  8. 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, Self-shrinkers & translators

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) 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)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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

  2. 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.

  3. 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.