For example consider the following question:
1.Let $\mathbb{B}^m$ be hyperbolic space and let $f
: \mathbb{B}^m \rightarrow \mathbb{B}^m$ be harmonic $K$-qc map.
Whether $f$ has critical points on $\mathbb{B}^m$ ?
2. Whether injective euclidean-harmonic map has critical points?
3. Suppose that $G \subset \mathbb{R}^n$ is a proper
subdomain, $f :G\rightarrow \mathbb{R}^n$ is harmonic $K$-qc and
$G'=f(G)$. Determine whether $f$ is a quasi-isometry (ie. bi-Lipschitz)
w.r.t. quasihyperbolic metrics on $G$ and $G'$.
In general, the questions are open.
The interested reader can find some information in Mateljevic paper: The
lower bound for the modulus of the derivatives and Jacobian of
harmonic injective mappings V3.
For example, Lewy's theorem is false
in dimensions higer than two (see Duren book p. 25-27 for Wood's
counterexample).
Consider the polynomial map from $\mathbb{R}^3$ to
$\mathbb{R}^3$ defined by $h(x,y,z)=(u,v,w)$, where
$$u = x^3 -3xz^2 + yz,\quad v=y-3xz, \quad w=z\,. $$
Since $z=w$, $u= x^3 + z v = x^3 + v w$, we find $x= \sqrt[3]{u-v w} $ and therefore $y=v+3xz= v +3xw= v +3w \sqrt[3]{u-v w} $, we conclude that $h$ is injective.
A calculation shows that $h$ has the Jacobian
$$ J_h(x,y,z)= 3x^2,$$
%\,
which vanishes on the plane $x=0$.
identifier 1805.02979 and is available at:
http://arxiv.org/abs/1805.02979
[1] Schwarz lemma and distortion for harmonic functions via length and area, 8 May 2018.
Authors: Miodrag Mateljevi\'c
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