Abstract
We formulate and apply a modified Lagrangian mean curvature flow to prescribe the Maslov form of Lagrangian immersions in ℂn. We prove longtime existence results and derive optimal results for curves.
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Abresch, U. and Langer, J. (1986) The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23(2), 175-196.
Angenent, S. (1991) On the formation of singularities in the curve shortening flow, J. Differential Geom. 33(3), 601-633.
Cao, H.-D. (1985) Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81, 359-372.
Gage, M. E. (1984) Curve shortening makes convex curves circular, Invent. Math. 76(2), 357-364.
Gage, M. and Hamilton, R. S. (1986) The heat equation shrinking convex plane curves, J. Differential Geom. 23(1), 69-96.
Grayson, M. A. (1989) The shape of a figure-eight under the curve shortening flow, Invent. Math. 96(1), 177-180.
Huisken, G. (1984) Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1), 237-266.
Morvan, J. M. (1981) Classe de Maslov d'une immersion lagrangienne et minimalité, CR Acad. Sci. Paris A 292, 633-636.
Oh, Y.-G. (1990) Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101(2), 501-519.
Schoen, R. and Wolfson, J. (1999) Minimizing volume among Lagrangian submanifolds, In: Differential Equations (La Pietra, Florence 1996), Proc. Sympos. Pure Math. 65, Amer. Math. Soc., Providence, RI, pp. 181-199.
Schoen, R. and Wolfson, J. (2000) Minimizing area among Lagrangian surfaces:The mapping problem, Preprint.
Smoczyk, K. (1999), Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 8(3), 247-258.
Strominger, A., Yau, S.-T. and Zaslow, E. (1996) Mirror symmetry is T-duality, Nuclear Phys. B 479, 243-259.
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Smoczyk, K. Prescribing the Maslov Form of Lagrangian Immersions. Geometriae Dedicata 91, 59–69 (2002). https://doi.org/10.1023/A:1016238817734
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DOI: https://doi.org/10.1023/A:1016238817734