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BY 4.0 license Open Access Published by De Gruyter Open Access October 22, 2020

Pseudo-holomorphic curves: A very quick overview

  • Gonçalo Oliveira EMAIL logo
From the journal Complex Manifolds


This is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length.

Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.


[1] Agmon, Shmuel. Lectures on exponential decay of solutions of second-order elliptic equations: Bounds on eigenfunctions of n-body Schrodinger operations.(mn-29). Vol. 29. Princeton University Press, 2014.10.2307/j.ctt13x1d8zSearch in Google Scholar

[2] Degeratu, A., Stern, M. Witten Spinors on Nonspin Manifolds. Commun. Math. Phys. 324, 301-350 (2013).10.1007/s00220-013-1804-0Search in Google Scholar

[3] S. Donaldson, Karen Uhlenbeck and the Calculus of Variations, March 2019, Notices of the American Mathematical Society 66(03):1 DOI: 10.1090/noti180610.1090/noti1806Search in Google Scholar

[4] M. Gromov, Pseudo holomorphic curves in symplectic manifolds. Inventiones mathematicae 82.2 (1985): 307-347.10.1007/BF01388806Search in Google Scholar

[5] Lin, Fang-Hua, and Tristan Riviere. Energy quantization for harmonic maps. Duke Mathematical Journal 111.1 (2002): 177-193.10.1215/S0012-7094-02-11116-8Search in Google Scholar

[6] D. McDuff, and D. Salamon, J-holomorphic curves and symplectic topology. Vol. 52. American Mathematical Soc., 2012.Search in Google Scholar

[7] Sacks, J., and K. Uhlenbeck. The existence of minimal immersions of two-spheres. Bulletin of the American Mathematical Society 83.5 (1977): 1033-1036.10.1090/S0002-9904-1977-14366-8Search in Google Scholar

[8] Wendl, Chris. Lectures on holomorphic curves in symplectic and contact geometry. arXiv preprint arXiv:1011.1690 (2010).Search in Google Scholar

Received: 2020-06-25
Accepted: 2020-10-11
Published Online: 2020-10-22

© 2020 Gonçalo Oliveira, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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