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Ancient solutions and translators of Lagrangian mean curvature flow

Jason D. Lotay, Felix Schulze, Gábor Székelyhidi

Abstract

Suppose that $\mathcal{M}$ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbb{C}^n$. We show that if $\mathcal{M}$ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then $\mathcal{M}$ is a translator. In particular in $\mathbb{C}^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

Ancient solutions and translators of Lagrangian mean curvature flow

Abstract

Suppose that is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in . We show that if has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then is a translator. In particular in , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.
Paper Structure (15 sections, 21 theorems, 118 equations)

This paper contains 15 sections, 21 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lagrangians in Cn
  4. Spacetime track
  5. Rescalings
  6. Set-up
  7. Translators
  8. The drift heat equation
  9. Limits of solutions of the heat equation
  10. Three annulus lemma
  11. Linking argument
  12. Approximate solutions of the heat equation
  13. The linking argument
  14. Proof of Main Theorem
  15. Smoothly immersed Brakke flows

Key Result

Theorem 1.1

Let $P_1,P_2\subset \mathbb{C}^n$ be Lagrangian subspaces which intersect along a line $\ell$ and have distinct Lagrangian angles. Let $\mathcal{M}$ be a smoothly immersed, ancient, Lagrangian Brakke flow in $\mathbb{C}^n$ with uniformly bounded area ratios. Assume further that $\mathcal{M}$ is exac

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 45 more