A twistor construction of Kahler submanifolds of a quaternionic Kahler manifold

摘要

A class of minimal almost complex submanifolds of a Riemannian manifold 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 K?hler submanifold is pluriminimal. In the case of a quaternionic K?hler manifold (M-tilde)~(4n) of non zero scalar curvature, in particular, when (M-tilde)~4 is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of K?hler submanifolds M~(2n) of maximal possible dimension 2n. More precisely, we prove that any such K?hler submanifold M~(2n) of is the projection of a holomorphic Legendrian submanifold of the twistor space (Z,H) of (M-tilde)~(4n), considered as a complex contact manifold with the natural holomorphic contact structure H is contained in TZ. Any Legendrian submanifold of the twistor space Zis defined by a generating holomorphic function. This is a natural generalization of Bryants construction of superminimal surfaces in S~4 = HP~1.