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

TL;DR

This paper establishes a bijective correspondence between superminimal surfaces in 4-manifolds and certain Lagrangian submanifolds in their twistor spaces, revealing new minimal Lagrangian submanifolds with geometric significance.

Contribution

It introduces a novel correspondence linking superminimal surfaces to Lagrangian submanifolds in twistor spaces, expanding understanding of their geometric properties.

Findings

Constructs Lagrangian submanifolds from superminimal surfaces.

Shows these Lagrangian submanifolds are minimal.

Provides examples in specific twistor spaces.

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.