Bounds for Jacobian of harmonic injective mappings in n-dimensional space

Abstract

Using normal family arguments, we show that the degree of the first nonzero homogenous polynomial in the expansion of n dimensional Euclidean harmonic K-quasiconformal mapping around an internal point is odd, and that such a map from the unit ball onto a bounded convex domain, with K < 3n-1, is co-Lipschitz. Also some generalizations of this result are given, as well as a generalization of Heinz?s lemma for harmonic quasiconformal maps in Rn and related results.