Proper holomorphic maps in harmonic map theory

We determine all proper holomorphic maps of balls $B_2 \to B_3$ admitting a $C^3$ extension up to the boundary of $B_2$ and whose boundary values $S^3\to S^5$ are subelliptic harmonic maps (in the sense of Jost and Xu inTrans. Am. Math. Soc. 350(11):4633–4649, 1998). A new numerical CR invariant, the CR degree of a CR map of spheres $S^{2n+1}\to S^{2N+1}$, 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 $\Phi \in P(2, 3)$ (cf. Alexander in Indiana Univ. Math. J. 26:137–146, 1977) is shown to be homotopically nontrivial, as a map of ${(z,w) \in S^3 : w +\overline{w} > 0}$ into $S^5\S^3$.