Are there static textures?

We consider harmonic maps from Minkowski space into the three-sphere. We are especially interested in solutions which are asymptotically constant, i.e., converge to the same value in all directions of spatial infinity. Physical three-space can then be compactified and topologically (but not metrically) identified with a three-sphere. Therefore for fixed time, the winding of the map is defined. We investigate whether static solutions with a nontrivial winding number exist. The answer which we can prove here is only partial: We show that within a certain family of maps no static solutions with a nonzero winding number exist. We discuss the existing static solutions in our family of maps. An extension to other maps or a proof that our family of maps is sufficiently general remains an open problem.