Skip to main content

Prescribing the Maslov Form of Lagrangian Immersions

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.

This is a preview of subscription content, access via your institution.

References

  • Abresch, U. and Langer, J. (1986) The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23(2), 175-196.

    Google Scholar 

  • Angenent, S. (1991) On the formation of singularities in the curve shortening flow, J. Differential Geom. 33(3), 601-633.

    Google Scholar 

  • Cao, H.-D. (1985) Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81, 359-372.

    Google Scholar 

  • Gage, M. E. (1984) Curve shortening makes convex curves circular, Invent. Math. 76(2), 357-364.

    Google Scholar 

  • Gage, M. and Hamilton, R. S. (1986) The heat equation shrinking convex plane curves, J. Differential Geom. 23(1), 69-96.

    Google Scholar 

  • Grayson, M. A. (1989) The shape of a figure-eight under the curve shortening flow, Invent. Math. 96(1), 177-180.

    Google Scholar 

  • Huisken, G. (1984) Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1), 237-266.

    Google Scholar 

  • Morvan, J. M. (1981) Classe de Maslov d'une immersion lagrangienne et minimalité, CR Acad. Sci. Paris A 292, 633-636.

    Google Scholar 

  • Oh, Y.-G. (1990) Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101(2), 501-519.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Strominger, A., Yau, S.-T. and Zaslow, E. (1996) Mirror symmetry is T-duality, Nuclear Phys. B 479, 243-259.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Smoczyk, K. Prescribing the Maslov Form of Lagrangian Immersions. Geometriae Dedicata 91, 59–69 (2002). https://doi.org/10.1023/A:1016238817734

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016238817734

  • Lagrangian
  • mean curvature flow
  • Maslov form