Abstract
In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) 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.
Similar content being viewed by others
References
Arezzo, C., Sun, J.: A variational characterization of J-holomorphic curves. J. Reine Angew. Math. 709, 171–200 (2015)
Arezzo, C., Sun, J.: A variational characterization of complex manifolds. Math. Ann. 366, 249–277 (2016)
Chern, S.S., Wolfson, J.: Minimal surfaces by moving frames. Am. J. Math. 105, 59–83 (1983)
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)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Inc, New York (1996)
Matsumoto, K.: A note on the differentiability of the distance function to regular submanifolds of Riemannian manifolds. Nihonkai Math. J. 3, 81–85 (1992)
McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence (2004)
Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 2(65), 391–404 (1957)
Nijenhuis, A., Woolf, W.B.: Some integration problems in almost-complex and complex manifolds. Ann. Math. 2(77), 424–489 (1963)
Pali, N.: Fonctions plurisousharmoniques et courants positifs de type \((1,1)\) sur une variété complex. Manuscripta Math. 118, 311–337 (2005)
Pliś, S.: Monge–Ampère operator on four dimensional almost complex manifolds. J. Geom. Anal. 26, 2503–2518 (2016)
Sun, J.: \(L_p\)-functional and \(J\)-holomorphic curves in almost Kähler manifolds. J. Math. Anal. Appl. 434, 1474–1488 (2016)
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
Corresponding author
Additional information
Supported by NSFC (China) Grants 11701226, 11501253, 11471145, 11401514, 11371309; Natural Science Foundation of Jiangsu Province BK20140525, BK20170519.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-017-9573-1