2019/10/20 by Reinier Storm, Storm, Reinier
Mathematics · #53C28 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C28 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1910.09033
arxiv created 2020/01/21 · arxiv updated 2020/01/22
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 ℂ P3 and \mathbbF1,2(ℂ3) with respect to both the Kähler structure as well as the nearly Kähler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.