Here's the setup: Let $U,V$ open $\subset \mathbb{R}^n$, and let $u:V\rightarrow \mathbb{R}$ be harmonic, and $v:U\rightarrow V$ be conformal, i.e. $v$ is $C^1$ and the Jacobian $J_v(x)$ is a scalar multiple of an orthogonal transformation for all [; x\in U ;].
I'm trying to prove $u\circ v$ is harmonic. [I've seen this stated as a fact in a few places without reference, namely here: http://en.wikipedia.org/wiki/Conformal_map#Uses , but maybe my hypotheses are slightly different and this is not true at all]
I've seen a proof that if $u$ is $C^2$ and $T$ is an orthogonal transformation then
$\Delta (u \circ T) = \Delta(u) \circ T$.
So I'm thinking that to show $u\circ v$ is harmonic, we can use the fact that $v$ acts locally as its Jacobian, which is an orthogonal transformation, and move the Laplacian onto $u$ and conclude $u\circ v$ is harmonic.
However, I'm having trouble making this idea precise. After glancing at my copy of baby Rudin, my hunch is to use the inverse function theorem or constant rank theorem, but I'm unsure how to apply those. Any suggestions?