Skip to main content
Log in

The existence of J-holomorphic curves in almost Hermitian manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (MgJF) be an almost Hermitian manifold and \(f:\Sigma \rightarrow M\) be an injective immersion. We prove that if the \(L_p\) functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arezzo, C., Sun, J.: A variational characterization of J-holomorphic curves. J. Reine Angew. Math. 709, 171–200 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Arezzo, C., Sun, J.: A variational characterization of complex manifolds. Math. Ann. 366, 249–277 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chern, S.S., Wolfson, J.: Minimal surfaces by moving frames. Am. J. Math. 105, 59–83 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demailly, J.-P.: Complex Analytic and Differential Geometry. Université de Grenoble I Institut Fourier, UMR 5582 du CNRS 38402 Saint-Martin d’Hères, France (2012)

  5. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Inc, New York (1996)

    MATH  Google Scholar 

  6. Matsumoto, K.: A note on the differentiability of the distance function to regular submanifolds of Riemannian manifolds. Nihonkai Math. J. 3, 81–85 (1992)

    MathSciNet  MATH  Google Scholar 

  7. McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence (2004)

    Book  MATH  Google Scholar 

  8. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 2(65), 391–404 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nijenhuis, A., Woolf, W.B.: Some integration problems in almost-complex and complex manifolds. Ann. Math. 2(77), 424–489 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pali, N.: Fonctions plurisousharmoniques et courants positifs de type \((1,1)\) sur une variété complex. Manuscripta Math. 118, 311–337 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pliś, S.: Monge–Ampère operator on four dimensional almost complex manifolds. J. Geom. Anal. 26, 2503–2518 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sun, J.: \(L_p\)-functional and \(J\)-holomorphic curves in almost Kähler manifolds. J. Math. Anal. Appl. 434, 1474–1488 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Professor Jun Sun for his patient discussion. The author also would like to thank the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qiang Tan.

Additional information

Supported by NSFC (China) Grants 11701226, 11501253, 11471145, 11401514, 11371309; Natural Science Foundation of Jiangsu Province BK20140525, BK20170519.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, Q. The existence of J-holomorphic curves in almost Hermitian manifolds. Ann Glob Anal Geom 53, 217–231 (2018). https://doi.org/10.1007/s10455-017-9573-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-017-9573-1

Keywords

Mathematics Subject Classification

Navigation