A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

Abstract

A class of minimal almost complex submanifolds of a Riemannian manifold \(\tilde{M}^{4n}\) with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kahler submanifold is defined. Any Kahler submanifold is pluriminimal. In the case of a quaternionic Kahler manifold \(\tilde{M}^{4n}\) of non zero scalar curvature, in particular, when \(\tilde{M}^4\) is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kahler submanifolds M2n of maximal possible dimension 2n. More precisely, we prove that any such Kahler submanifold M2n of \(\tilde{M}^{4n}\) is the projection of a holomorphic Legendrian submanifold \(L^{2n} \subset\mathcal{Z}\) of the twistor space \((\mathcal{Z},\mathcal{H})\) of \(\tilde{M}^{4n}\), considered as a complex contact manifold with the natural holomorphic contact structure \(\mathcal{H} \subset{T}\mathcal{Z}\). Any Legendrian submanifold of the twistor space \(\mathcal{Z}\) is defined by a generating holomorphic function. This is a natural generalization of Bryant’s construction of superminimal surfaces in S4=ℍP1.