Proper holomorphic maps in harmonic map theory

We determine all proper holomorphic maps of balls B-2 -> B-3 admitting a C-3 extension up to the boundary of B-2 and whose boundary values S-3 -> S-5 are subelliptic harmonic maps (in the sense of Jost and Xu in Trans Am Math Soc 350(11): 4633-4649, 1998). A new numerical CR invariant, the CR degree of a CR map of spheres S2n+ 1 -> S2N+1, is introduced and used to distinguish among the spherical equivalence classes in Faran's list P*(2, 3) (cf. Faran in Invent Math 68: 441-475, 1982). As an application, the boundary values phi of Alexander's map Phi is an element of P(2, 3) (cf. Alexander in Indiana Univ Math J 26: 137-146, 1977) is shown to be homotopically nontrivial, as a map of {(z, w) is an element of S-3 : w + (w) over bar > 0} into S-5\S-3.