As on arXiv
Abstract
We study mean curvature flow of Lagrangians in that are cohomogeneity-one with respect to a compact Lie group acting linearly on . Each such Lagrangian necessarily lies in a level set of the standard moment map , and mean curvature flow preserves this containment. We classify all cohomogeneity-one self-similarly shrinking, expanding and translating solutions to the flow, as well as cohomogeneity-one smooth special Lagrangians lying in . Restricting to the case of almost-calibrated flows in the zero level set , we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in , we show it occurs as the Type II blowup model of a Lagrangian MCF singularity. Throughout, we give explicit examples of suitable group actions, including a complete list in the case of simple. This yields infinitely many new examples of shrinking and expanding solitons for Lagrangian MCF, as well as infinitely many new singularity models.
Comments: 53 pages, 3 figures; v2: minor changes, corrected erroneous proofs of Proposition 3.13 and Corollary 3.15
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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.
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Infinite-Time Singularities of the Lagrangian Mean Curvature Flow
Topics Lagrangian mean curvature flow, Singularities · Same author Albert Wood · Shared terms “special Lagrangian”, “time singularity”
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Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics
Topics Lagrangian mean curvature flow, Homogeneous spaces, Singularities · Shared terms “special Lagrangian”, “cohomogeneity”, “almost calibrated”
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Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
Topics Lagrangian mean curvature flow, Homogeneous spaces, Singularities · Same author Albert Wood · Shared terms “cohomogeneity”, “time singularity”, “finite time”
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Ancient solutions and translators of Lagrangian mean curvature flow
Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “almost calibrated”, “special Lagrangian”
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Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow
Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Ricci solitons
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Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class
Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Singularities · Shared terms “almost calibrated”
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Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II
Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “special Lagrangian”
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Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class
Topics Lagrangian mean curvature flow, Self-shrinkers & translators, Singularities
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6C.8Monotonicity, ε-Regularity and Tangent Cones
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.