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The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ2

  • Jingyi Chen EMAIL logo and John Man Shun Ma
Published/Copyright: March 10, 2016
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 743

Abstract

Let Fn : (Σ, hn) →ℂ2 be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics {hn} converges smoothly to a Riemannian metric h. We show that a subsequence of {Fn} converges smoothly to a branched conformally immersed Lagrangian self-shrinker F∞ : (Σ, h) →ℂ2. When the area bound is less than 16π, the limit F∞ is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence hn→h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: RGPIN 203199-1

Funding statement: The first author is partially supported by an NSERC grant (RGPIN 203199-1).

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Received: 2014-11-05
Revised: 2015-10-27
Published Online: 2016-03-10
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the extension theorem of Hwang and Mok
  3. Tilting theory via stable homotopy theory
  4. Balanced line bundles on Fano varieties
  5. Symmetric differentials and variations of Hodge structures
  6. Multi-bump solutions of -Δn = K(x)u(n+2)/(n-2) on lattices in ℝn
  7. Asymptotic estimates on the von Neumann inequality for homogeneous polynomials
  8. The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ2
  9. Quintic threefolds and Fano elevenfolds
  10. The Asaeda–Haagerup fusion categories
https://doi.org/10.1515/crelle-2015-0110

Articles in the same Issue

  1. Frontmatter
  2. On the extension theorem of Hwang and Mok
  3. Tilting theory via stable homotopy theory
  4. Balanced line bundles on Fano varieties
  5. Symmetric differentials and variations of Hodge structures
  6. Multi-bump solutions of -Δn = K(x)u(n+2)/(n-2) on lattices in ℝn
  7. Asymptotic estimates on the von Neumann inequality for homogeneous polynomials
  8. The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ2
  9. Quintic threefolds and Fano elevenfolds
  10. The Asaeda–Haagerup fusion categories
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