Abstract
We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in \({\mathbb{c}}^n\) .
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Anciaux, H.: Construction of Lagrangian self-similar solutions to the mean curvature flow in \({\mathbb{c}}^n\) , Preprint (2004)
Audin, M., Lalonde, F., Polterovich, L.: Symplectic rigidity: Lagrangian submanifolds, In: Holomorphic curves in symplectic geometry, vol. 117 of Progr. Math., pp 271–321. Birkhäuser, Basel (1994)
Chekanov Y. (1996). Lagrangian tori in a symplectic vector space and global symplectomorphisms. Math. Z. 223: 547–559
Cieliebak, K., Goldstein, E.: A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions, 5p, (Preprint) (2004)
Eliashberg, Y.: Topology of 2-knots in R 4 and symplectic geometry, In: The Floer memorial volume, vol 133 of Progr. Math., pp 335–353. Birkhäuser, Basel (1995)
Eliashberg Y. and Polterovich L. (1996). Local Lagrangian 2-knots are trivial. Ann. Math. 144(2): 61–76
Eliashberg, Y., Polterovich, L.: The problem of Lagrangian knots in four-manifolds. In: Geometric topology (Athens, GA, 1993), vol. 2 of AMS/IP, pp 313–327. Stud. Adv. Math. Amer. Math. Soc., Providence (1997)
Groh, K.: Contraction of equivariant Lagrangian embeddings under the mean curvature flow. University of Hannover, preprint (2006)
Gromov M. (1985). Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82: 307–347
Huisken G. (1990). Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1): 285–299
Morvan J.-M. (1981). Classe de Maslov d’une immersion lagrangienne et minimalité. C. R. Acad. Sci. Paris Sér. I Math. 292: 633–636
Neves, A.: Singularities of Lagrangian mean curvature flow. Preprint (2005)
Neves, A.: Singularities of Lagrangian mean curvature flow: Monotone case. preprint, arXiv.org math.DG/0608401 (2006)
Oh Y.-G. (1993). Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I. Comm. Pure Appl. Math. 46: 949–993
Smoczyk K. (2005). Self-shrinkers of the mean curvature flow in arbitrary codimension. IMRN 48: 2983–3004
Smoczyk K. (2002). Angle theorems for the Lagrangian mean curvature flow. Math. Z. 240: 849–883
Smoczyk K. (1996). Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature. Calc. Var. 4: 155–170
Smoczyk K. and Wang M.-T. (2002). Mean curvature flows for Lagrangian submanifolds with convex potentials. J. Differ. Geom. 62: 243–257
Zehmisch, K.: Isotopies of Lagrangian tori in symplectic vector spaces—Part III. Leipzig, January 31, preprint (2005)
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Supported by DFG, priority program SPP 1154, SM 78/1-1, SCHW 892/1-1.
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Groh, K., Schwarz, M., Smoczyk, K. et al. Mean curvature flow of monotone Lagrangian submanifolds. Math. Z. 257, 295–327 (2007). https://doi.org/10.1007/s00209-007-0126-3
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DOI: https://doi.org/10.1007/s00209-007-0126-3