Abstract
By employing a nice adapted frame we prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kähler manifold \(S^3\times S^3\). The proof uses a local correspondence between almost complex surfaces in \(S^3\times S^3\) and surfaces in \(\mathbb {R}^3\) that satisfy the Wente H-surface equation. Furthermore we give a complete classification of flat almost complex surfaces in the homogeneous nearly Kähler \(S^3\times S^3\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bolton, J., Dillen, F., Dioos, B., Vrancken, L.: Almost complex surfaces in the nearly Kähler \(S^3\times S^3\). Tohoku Math. J. 67, 1–17 (2015)
Bolton, J., Vrancken, L., Woodward, L.M.: On almost complex curves in the nearly Kähler 6-sphere. Q. J. Math. Oxf. Ser. (2) 45, 407–427 (1994)
Bolton, J., Pedit, F., Woodward, L.: Minimal surfaces and the affine Toda field model. J. Reine Angew. Math. 459, 119–150 (1995)
Bryant, R.L.: Submanifolds and special structures on the octonians. J. Differ. Geom. 17, 185–232 (1982)
Butruille, J.-B.: Homogeneous nearly Kähler manifolds. In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry. RMA Lectures in Mathematics and Theoretical Physics, vol. 16, pp. 399–423. Eur. Math. Soc., Zürich (2010)
Dillen, F., Verstraelen, L., Vrancken, L.: Almost complex submanifolds of a 6-dimensional sphere II. Kodai Math. J. 10, 161–171 (1987)
Foscolo, L., Haskins, M.: New \(G_2\) holonomy cones and exotic nearly Kähler structures on \(S^6\) and \(S^3\times S^3\). Ann. Math. 185(1), 59–130 (2017)
Gray, A.: Almost complex submanifolds of the six sphere. Proc. Am. Math. Soc. 20, 272–276 (1969)
Gray, A., Hervella, L.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 123, 35–58 (1980)
Podestà, F., Spiro, A.: \(6\)-Dimensional nearly Kähler manifolds of cohomogeneity one. J. Geom. Phys. 60, 156–164 (2010)
Sekigawa, K.: Almost complex submanifolds of a 6-dimensional sphere. Kodai Math. J. 6, 147–185 (1983)
Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)
Xu, F.: Pseudo-holomorphic curves in nearly Kähler \(\mathbf{CP}^3\). Differ. Geom. Appl. 28, 107–120 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Tsinghua University–K. U. Leuven Bilateral Scientific Cooperation Fund. H. Li was supported by National Natural Foundation of China (Grant No. 11271214) and H. Ma was supported by National Natural Foundation of China (Grant No. 11271213).
Rights and permissions
About this article
Cite this article
Dioos, B., Li, H., Ma, H. et al. Flat Almost Complex Surfaces in the Homogeneous Nearly Kähler \({\varvec{S}}^\mathbf{3}\varvec{\times } {\varvec{S}}^\mathbf{3}\). Results Math 73, 38 (2018). https://doi.org/10.1007/s00025-018-0784-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0784-y