Geometriae Dedicata

, Volume 91, Issue 1, pp 59–69 | Cite as

Prescribing the Maslov Form of Lagrangian Immersions

  • K. Smoczyk


We formulate and apply a modified Lagrangian mean curvature flow to prescribe the Maslov form of Lagrangian immersions in ℂ n . We prove longtime existence results and derive optimal results for curves.

Lagrangian mean curvature flow Maslov form 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abresch, U. and Langer, J. (1986) The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23(2), 175-196.Google Scholar
  2. Angenent, S. (1991) On the formation of singularities in the curve shortening flow, J. Differential Geom. 33(3), 601-633.Google Scholar
  3. Cao, H.-D. (1985) Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81, 359-372.Google Scholar
  4. Gage, M. E. (1984) Curve shortening makes convex curves circular, Invent. Math. 76(2), 357-364.Google Scholar
  5. Gage, M. and Hamilton, R. S. (1986) The heat equation shrinking convex plane curves, J. Differential Geom. 23(1), 69-96.Google Scholar
  6. Grayson, M. A. (1989) The shape of a figure-eight under the curve shortening flow, Invent. Math. 96(1), 177-180.Google Scholar
  7. Huisken, G. (1984) Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1), 237-266.Google Scholar
  8. Morvan, J. M. (1981) Classe de Maslov d'une immersion lagrangienne et minimalité, CR Acad. Sci. Paris A 292, 633-636.Google Scholar
  9. Oh, Y.-G. (1990) Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101(2), 501-519.Google Scholar
  10. 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
  11. Schoen, R. and Wolfson, J. (2000) Minimizing area among Lagrangian surfaces:The mapping problem, Preprint.Google Scholar
  12. Smoczyk, K. (1999), Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 8(3), 247-258.Google Scholar
  13. Strominger, A., Yau, S.-T. and Zaslow, E. (1996) Mirror symmetry is T-duality, Nuclear Phys. B 479, 243-259.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • K. Smoczyk
    • 1
  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations