The Euler-Lagrange equations for extremal holomorphic mappings of the unit disk (Q1065974)

The author considers extremal problems for the class H(\(\Delta\),D) of holomorphic mappings f of the planar unit disk \(\Delta\) into a domain D of \({\mathbb{C}}^ n\). As is well known, the supremum of \(\| f'(0)\|\) over \(f\in H(\Delta,D)\) gives the infinitesimal Royden-Kobayashi metric at the point \(z=f(0)\) which is ''distance-decreasing'', and hence is biholomorphically invariant. In particular, the extremal mappings are invariant under biholomorphic transformations, and the author conjectures that their boundary values coincide with the Moser chains. The proof of the last conjecture requires the knowledge that the boundary values lie on the boundary of the domain. This property is proved for large classes of functionals. The extremals are computed via the Euler-Lagrange equations, which in this paper are derived in the case of pseudoconvex domains. Some of the results are similar to those of Lempert, who used different methods.