Skip to main content
SpringerLink
Log in

Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein

  • Conference paper
  • First Online:
Book cover Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.

Mathematics Subject Classification (2010) 53C44.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Dazord, Sur la géométrie des sous-fibrés et des feuilletages lagrangiens, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 465–480 (1982).

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12. Oxford University Press, Oxford, 2007.

    Google Scholar 

  4. D. D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces, Ann. Global Anal. Geom. 25 (2004), no. 4, 301–352.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. D. Joyce, Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications, J. Differential Geom. 63 (2003), 279–347.

    MATH  MathSciNet  Google Scholar 

  6. O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967.

    Google Scholar 

  7. D. McDuff, D. Salamon, Introduction to symplectic topology, Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.

    Google Scholar 

  8. D. McDuff, D. Salamon, J-holomorphic curves and quantum cohomology, University Lecture Series, 6. American Mathematical Society, Providence, RI, 1994.

    Google Scholar 

  9. R. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), no. 4, 705–747.

    MATH  MathSciNet  Google Scholar 

  10. K. Smoczyk, Longtime existence of the Lagrangian mean curvature flow, Calc. Var. 20, 25–46 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  11. K. Smoczyk, A canonical way to deform a Lagrangian submanifold, arXiv:dg-ga/9605005 (1996).

    Google Scholar 

  12. K. Smoczyk, M.-T. Wang, Generalized Lagrangian mean curvature flows in symplectic manifolds, arXiv:0910.2667 (2009).

    Google Scholar 

  13. K. Smoczyk, M.-T. Wang, Mean curvature flows of Lagrangians submanifolds with convex potentials, J. Differential Geom. 62 (2002), no. 2, 243–257.

    MATH  MathSciNet  Google Scholar 

  14. A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), no. 1–2, 243–259.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. P. Thomas, S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113.

    MATH  MathSciNet  Google Scholar 

  16. G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986), 629–646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.

    Google Scholar 

  17. A. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), no. 2, 325–346.

    Article  MATH  MathSciNet  Google Scholar 

  18. M.-T. Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential Geom. 57 (2001), no. 2, 301–338.

    MATH  MathSciNet  Google Scholar 

  19. J. Wolfson, Lagrangian homology classes without regular minimizers, J. Differential Geom. 71 (2005), no. 2,307–313.

    MATH  MathSciNet  Google Scholar 

  20. S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St-Giles’, OX1 3LB, Oxford, UK

    Tapio Behrndt

Authors
  1. Tapio Behrndt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tapio Behrndt .

Editor information

Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Behrndt, T. (2011). Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation