Skip to main content

Minimal lagrangian submanifolds of Kähler-einstein manifolds

Part of the Lecture Notes in Mathematics book series (LNM,volume 1255)

Abstract

An interesting class of submanifolds of a Kähler manifold M2n is the class of submanifolds Nn ⊑ M2n which are minimal with respect to the metric on M2n and are Lagrangian with respect to the symplectic form on M2n. A general Kähler manifold will not have any of these submanifolds. However, in this paper, we show that if the metric on M2n is also Einstein, then these minimal Lagrangian submanifolds exist in abundance, at least locally. We give a precise description of this "generality" in terms of Cartan-Kähler theory and relate these submanifolds to the calibrated geometries of Harvey and Lawson and to maximal real structures on algebraic varieties.

Keywords

  • Complex Manifold
  • Differential System
  • Algebraic Variety
  • Lagrangian Submanifold
  • Frame Field

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. S.S. Chern, Complex Manifolds without Potential Theory, 2nd edition, Springer-Verlag, 1979

    Google Scholar 

  2. S.S. Chern, et al., Essays in Exterior Differential Systems, to appear.

    Google Scholar 

  3. Dennis de Turck and Jerry Kazdan, Some Regularity Theorems in Riemannian Geometry, Ann. Scient. Ec. Norm. Sup., 4e série, t. 14, 1981, pp 249–260

    MathSciNet  MATH  Google Scholar 

  4. Reese Harvey and Blaine Lawson, Calibrated Geometries, Acta Mathematica, v. 148 (1982), pp 47–157

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Alan Weinstein, Lectures on Symplectic Manifolds, CMBS Series in Mathematics, no. 29, AMS, 1977

    Google Scholar 

  6. S.T. Yau, Survey on Partial Differential Equations in Differential Geometry, Annals of Math. Studies, no. 102, Princeton University Press, 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Bryant, R.L. (1987). Minimal lagrangian submanifolds of Kähler-einstein manifolds. In: Gu, C., Berger, M., Bryant, R.L. (eds) Differential Geometry and Differential Equations. Lecture Notes in Mathematics, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077676

Download citation

  • DOI: https://doi.org/10.1007/BFb0077676

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17849-1

  • Online ISBN: 978-3-540-47883-6

  • eBook Packages: Springer Book Archive