Paper · Jan 2, 2019

-minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

Wei-Bo Su

math.DGarXiv:1901.00259
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Abstract

The aim of this paper is to study variational properties for -minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derive second variation formula for -minimal Lagrangians as a generalization of Chen and Oh's formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to -volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF, and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

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Submitted
Jan 2, 2019
Subjects
math.DG

Comments: 19 pages

Mean curvature flowLagrangian mean curvature flowRicci solitonsKähler–Ricci solitonsSelf-shrinkers & translatorsKähler geometryStability & uniquenessBoundaries & non-compact

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 Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow (Toru Kajigaya and Keita Kunikawa, 2017)5Kajigaya 2017 On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow (Wei-Bo Su, 2020)1Su 2020 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)3Su 2024 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)6Lee 2024 Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow (Maxwell Stolarski and Wei-Bo Su, 2026)4Stolarski 2026 Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics (Maxwell Stolarski and Wei-Bo Su, 2026)8Stolarski 2026 Almost isoclinic Lagrangian submanifolds (Tang-Kai Lee and Mu-Tao Wang, 2026)2Lee 2026 Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature (Maxwell Stolarski and Wei-Bo Su, 2026)7Stolarski 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow

    Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Wei-Bo Su · Shared terms “translating solitons”, “special Lagrangian”

  2. Almost isoclinic Lagrangian submanifolds

    Tang-Kai Lee and Mu-Tao WangWider flows

    Topics Lagrangian mean curvature flow, Stability & uniqueness · Shared terms “Lagrangian submanifold”, “minimal Lagrangian”

  3. Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

    Wei-Bo Su, Chung-Jun Tsai and Albert Wood

    Topics Lagrangian mean curvature flow · Same author Wei-Bo Su · Shared terms “special Lagrangian”

  4. Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Wei-Bo Su · Shared terms “special Lagrangian”

  5. Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

    Toru Kajigaya and Keita KunikawaWider flows

    Topics Lagrangian mean curvature flow, Kähler geometry, Stability & uniqueness · Shared terms “Lagrangian submanifold”, “minimal Lagrangian”, “variational”

  6. Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature

    Ping-Hung Lee and Chung-Jun TsaiWider flows

    Topics Lagrangian mean curvature flow, Kähler geometry, Stability & uniqueness · Shared terms “minimal Lagrangian”

  7. Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow · Same author Wei-Bo Su

  8. Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow · Same author Wei-Bo Su · Shared terms “special Lagrangian”, “calibrated”

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) 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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000)

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.