A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces

DOI10.1016/J.DIFGEO.2020.101669arXiv1910.09033WikidataQ115354723 ScholiaQ115354723MaRDI QIDQ6327544

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Publication date: 20 October 2019

Abstract: In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian $4$-manifold and particular Lagrangian submanifolds of the twistor space over the $4$-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces $mathbbC{P}^3$ and $mathbb{F}_{1,2}(mathbb{C}^3)$ with respect to both the K"{a}hler structure as well as the nearly K"{a}hler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.

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