Geometry & Topology

Uniqueness of Lagrangian self-expanders

Jason D Lotay and André Neves

Full-text: Open access

Abstract

We show that zero-Maslov class Lagrangian self-expanders in n that are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.

Article information

Source
Geom. Topol., Volume 17, Number 5 (2013), 2689-2729.

Dates
Received: 14 August 2012
Accepted: 29 April 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732686

Digital Object Identifier
doi:10.2140/gt.2013.17.2689

Mathematical Reviews number (MathSciNet)
MR3190297

Zentralblatt MATH identifier
1304.53067

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Keywords
Lagrangian mean curvature flow self-expanders uniqueness

Citation

Lotay, Jason D; Neves, André. Uniqueness of Lagrangian self-expanders. Geom. Topol. 17 (2013), no. 5, 2689--2729. doi:10.2140/gt.2013.17.2689. https://projecteuclid.org/euclid.gt/1513732686


Export citation

References

  • H Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\mathbb{C}^n$, Geom. Dedicata 120 (2006) 37–48
  • M Cantor, Sobolev inequalities for Riemannian bundles, from: “Differential geometry, Part 2”, (S S Chern, R Osserman, editors), Amer. Math. Soc. (1975) 171–184
  • I Castro, A M Lerma, Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane, Proc. Amer. Math. Soc. 138 (2010) 1821–1832
  • A Chau, J Chen, W He, Entire self-similar solutions to Lagrangian mean curvature flow
  • K Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications 57, Birkhäuser, Boston, MA (2004)
  • G Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285–299
  • T Ilmanen, Singularities of mean curvature flow of surfaces Available at \setbox0\makeatletter\@url http://www.math.ethz.ch/~ilmanen/papers/sing.ps {\unhbox0
  • T Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994) 90
  • D Joyce, Special Lagrangian submanifolds with isolated conical singularities, II: Moduli spaces, Ann. Global Anal. Geom. 25 (2004) 301–352
  • D Joyce, Y-I Lee, M-P Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010) 127–161
  • N V Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics 12, Amer. Math. Soc. (1996)
  • S Lang, Real and functional analysis, 3rd edition, Graduate Texts in Mathematics 142, Springer, New York (1993)
  • Y-I Lee, M-T Wang, Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows, J. Differential Geom. 83 (2009) 27–42
  • Y-I Lee, M-T Wang, Hamiltonian stationary cones and self-similar solutions in higher dimension, Trans. Amer. Math. Soc. 362 (2010) 1491–1503
  • H Nakahara, Some examples of self-similar solutions and translating solitons for mean curvature flow
  • A Neves, Singularities of Lagrangian mean curvature flow: Zero-Maslov class case, Invent. Math. 168 (2007) 449–484
  • A Neves, Recent progress on singularities of Lagrangian mean curvature flow, from: “Surveys in geometric analysis and relativity”, (H L Bray, W P Minicozzi II, editors), Adv. Lect. Math. 20, International Press (2011) 413–438
  • A Neves, Finite time singularities for Lagrangian mean curvature flow, Ann. of Math. 177 (2013) 1029–1076
  • A Neves, G Tian, Translating solutions to Lagrangian mean curvature flow, Trans. Amer. Math. Soc. 365 (2013) 5655–5680
  • A Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics 1764, Springer, Berlin (2001)
  • L Simon, Lectures on geometric measure theory, Proc. Centre Math. Analysis 3, Australian Nat. Univ. Centre Math. Analysis, Canberra (1983)
  • R P Thomas, S-T Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002) 1075–1113
  • B White, A local regularity theorem for mean curvature flow, Ann. of Math. 161 (2005) 1487–1519