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

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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 H 1(L,ℤ).

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References

  1. Anciaux, H.: Mean curvature flow and self-similar submanifolds. Séminaire de Théorie Spectrale et Gémométrie, vol. 21, pp. 43–53. (2002–2003)

  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. Chen, J., Li, J.: Singularity of mean curvature flow of Lagrangian submanifolds. Invent. Math. 156, 25–51 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, J., Li, J., Tian, G.: Two-dimensional graphs moving by mean curvature flow. Acta Math. Sin. 18, 209–224 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ecker, K.: Regularity theory for mean curvature flow. in Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser, Boston, MA (2004)

  6. Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130(2), 453–471 (1989)

    MathSciNet  Google Scholar 

  7. Harvey, R., Lawson, H.B.: H. Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)

    MATH  MathSciNet  Google Scholar 

  9. Ilmanen, T.: Singularities of Mean Curvature Flow of Surfaces. Preprint.

  10. Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National University

  11. Schoen, R., Wolfson, J.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 58, 1–86 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Smoczyk, K.: A canonical way to deform a Lagrangian submanifold. Preprint.

  13. Smoczyk, K.: Harnack inequality for the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 8, 247–258 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Smoczyk, K.: Angle theorems for the Lagrangian mean curvature flow. Math. Z. 240, 849–883 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Smoczyk, K.: Longtime existence of the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 20, 25–46 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Smoczyk, K., Wang, M.-T.: Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differ. Geom. 62, 243–257 (2002)

    MATH  MathSciNet  Google Scholar 

  17. Tsui, M.-P., Wang, M.-T.: Mean curvature flows and isotopy of maps between spheres. Commun. Pure Appl. Math. 57, 1110–1126 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang, M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57, 301–338 (2001)

    MATH  Google Scholar 

  19. Wang, M.-T.: Deforming area preserving diffeomorphism of surfaces by mean curvature flow. Math. Res. Lett. 8, 651–661 (2001)

    MATH  MathSciNet  Google Scholar 

  20. Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148, 525–543 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang, M.-T.: Gauss maps of the mean curvature flow. Math. Res. Lett. 10, 287–299 (2003)

    MATH  MathSciNet  Google Scholar 

  22. White, B.: A local regularity theorem for mean curvature flow. Ann. Math. 161, 1487–1519 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto Superior Técnico, Lisbon, Portugal

    André Neves

  2. Mathematics Department, Stanford University, Stanford, CA, 94305, USA

    André Neves

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  1. André Neves
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Correspondence to André Neves.

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Neves, A. Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. math. 168, 449–484 (2007). https://doi.org/10.1007/s00222-007-0036-3

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  • DOI: https://doi.org/10.1007/s00222-007-0036-3

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