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

TL;DR

This paper establishes bounds on the Jacobian of harmonic quasiconformal mappings in n-dimensional space, showing conditions under which such maps are co-Lipschitz and providing generalizations of classical results.

Contribution

It introduces new bounds for the Jacobian of harmonic quasiconformal maps in higher dimensions and extends classical lemmas to this setting.

Findings

Degree of the first nonzero homogeneous polynomial is odd

Harmonic K-quasiconformal maps from the unit ball are co-Lipschitz for K<3^{n-1}

Generalizations of Heinz's lemma for harmonic quasiconformal maps

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 $K< 3^{n-1}$, 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 $\mathbb R^n$ and related results.