Are there static textures? art. no. 125007

We consider harmonic maps from Minkowski space into the three-sphere. We ar e especially interested in solutions which are asymptotically constant, i.e ., converge to the same value in all directions of spatial infinity. Physic al three-space can then be compactified and topologically (but not metrical ly) identified with a three-sphere. Therefore for fixed time, the winding o f the map is defined. We investigate whether static solutions with a nontri vial winding number exist. The answer which we can prove here is only parti al: 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 o f maps is sufficiently general remains an open problem. [S0556-2821(99)0461 2-3].