Twistorial construction of minimal hypersurfaces (Q2925881)

Finding examples of minimal submanifolds is an old and interesting problem. This paper presents a beautiful construction of minimal submanifolds by exploiting Penrose's twistor space construction, i.e., consider the hypersurface (in the space of all compatible almost complex structures on a Riemannian manifold \(X\)) consisting of all compatible almost complex structures commuting with a given one on \(X\). This hypersurface is minimal under some conditions. Indeed, the main results are of the flavour ``If we consider a four manifold of a certain type, then easy-to-verify geometric conditions on it lead to a natural minimal hypersurface of its twistor space.'' For instance, if \(X\) is a complex surface, the corresponding hypersurface is minimal if and only if a certain 2-form on \(X\) is of type \((1,1)\). NEWLINENEWLINENEWLINEThe author presents some concrete examples as well. Namely, the aforementioned hypersurface in the twistor space of generalized Hopf surfaces and certain Kodaira surfaces.