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

金沢大学人間社会研究域学校教育系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 hyperkähler 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\u27s theorem for a constant mean curvature surface of genus zero in threedimensional 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. © 2010 American Mathematical Society