Inventiones mathematicae

, Volume 168, Issue 3, pp 449–484 | Cite as

Singularities of Lagrangian mean curvature flow: zero-Maslov class case

Article

Abstract

We study singularities of Lagrangian mean curvature flow in ℂn when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian isotopic to a plane that, nevertheless, develop finite time singularities under mean curvature flow.

We then prove two theorems regarding the tangent flow at a singularity when the initial condition is a zero-Maslov class Lagrangian. The first one (Theorem A) states that that the rescaled flow at a singularity converges weakly to a finite union of area-minimizing Lagrangian cones. The second theorem (Theorem B) states that, under the additional assumptions that the initial condition is an almost-calibrated and rational Lagrangian, connected components of the rescaled flow converges to a single area-minimizing Lagrangian cone, as opposed to a possible non-area-minimizing union of area-minimizing Lagrangian cones. The latter condition is dense for Lagrangians with finitely generated H1(L,ℤ).

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References

  1. 1.
    Anciaux, H.: Mean curvature flow and self-similar submanifolds. Séminaire de Théorie Spectrale et Gémométrie, vol. 21, pp. 43–53. (2002–2003)Google Scholar
  2. 2.
    Angenent, S.: Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions. Ann. Math. 133(2), 171–215 (1991)Google Scholar
  3. 3.
    Chen, J., Li, J.: Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math. 156, 25–51 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, J., Li, J., Tian, G.: Two-dimensional graphs moving by mean curvature flow. Acta Math. Sin. 18, 209–224 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ecker, K.: Regularity theory for mean curvature flow. in Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser, Boston, MA (2004)Google Scholar
  6. 6.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130(2), 453–471 (1989)MathSciNetGoogle Scholar
  7. 7.
    Harvey, R., Lawson, H.B.: H. Calibrated geometries. Acta Math. 148, 47–157 (1982)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)MATHMathSciNetGoogle Scholar
  9. 9.
    Ilmanen, T.: Singularities of Mean Curvature Flow of Surfaces. Preprint.Google Scholar
  10. 10.
    Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National UniversityGoogle Scholar
  11. 11.
    Schoen, R., Wolfson, J.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 58, 1–86 (2001)MATHMathSciNetGoogle Scholar
  12. 12.
    Smoczyk, K.: A canonical way to deform a Lagrangian submanifold. Preprint.Google Scholar
  13. 13.
    Smoczyk, K.: Harnack inequality for the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 8, 247–258 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Smoczyk, K.: Angle theorems for the Lagrangian mean curvature flow. Math. Z. 240, 849–883 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Smoczyk, K.: Longtime existence of the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 20, 25–46 (2004)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Smoczyk, K., Wang, M.-T.: Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differ. Geom. 62, 243–257 (2002)MATHMathSciNetGoogle Scholar
  17. 17.
    Tsui, M.-P., Wang, M.-T.: Mean curvature flows and isotopy of maps between spheres. Commun. Pure Appl. Math. 57, 1110–1126 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wang, M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57, 301–338 (2001)MATHGoogle Scholar
  19. 19.
    Wang, M.-T.: Deforming area preserving diffeomorphism of surfaces by mean curvature flow. Math. Res. Lett. 8, 651–661 (2001)MATHMathSciNetGoogle Scholar
  20. 20.
    Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148, 525–543 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang, M.-T.: Gauss maps of the mean curvature flow. Math. Res. Lett. 10, 287–299 (2003)MATHMathSciNetGoogle Scholar
  22. 22.
    White, B.: A local regularity theorem for mean curvature flow. Ann. Math. 161, 1487–1519 (2005)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Instituto Superior TécnicoLisbonPortugal
  2. 2.Mathematics DepartmentStanford UniversityStanfordUSA

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