Harmonicity and minimality of vector fields and distributions on locally conformal Kähler and hyperkähler manifolds

Abstract

We show that on any locally conformal Kähler (l.c.K.) manifold $(M,J,g)$ with parallel Lee form the unit anti-Lee vector field is harmonic and minimal and determines a harmonic map into the unit tangent bundle. Moreover, the canonical distribution locally generated by the Lee and anti-Lee vector fields is also harmonic and minimal when seen as a map from $(M,g)$ with values in the Grassmannian $G^{or}_2(M)$ endowed with the Sasaki metric. As a particular case of l.c.K. manifolds, we look at locally conformal hyperkähler manifolds and show that, if the Lee form is parallel (that is always the case if the manifold is compact), the naturally associated three- and four-dimensional distributions are harmonic and minimal when regarded as maps with values in appropriate Grassmannians. As for l.c.K. manifolds without parallel Lee form, we consider the Tricerri metric on an Inoue surface and prove that the unit Lee and anti-Lee vector fields are harmonic and minimal and the canonical distribution is critical for the energy functional, but when seen as a map with values in $G^{or}_2(M)$ it is minimal, but not harmonic.