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

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 $\mathbb{C} P^3$ 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.