Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections

Abstract

We determine surfaces of genus zero in self-dual Einstein manifolds whose twistor lifts are harmonic sections. We apply our main theorem to the case of four-dimensional hyperkahler manifolds. As a corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is a harmonic section. In particular, if the mean curvature vector field is parallel with respect to the normal connection, then the surface is totally umbilic. Thus, our main theorem is a generalization of Hopf's theorem for a constant mean curvature surface of genus zero in three-dimensional Euclidean space. Moreover, we can also see that a Lagrangian surface of genus zero in the complex Euclidean plane with conformal Maslov form is the Whitney sphere.