Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is (weakly) conformal and $f:\mathcal{N}\rightarrow\mathcal{P}$ is harmonic, then $f\circ\phi:\mathcal{M}\rightarrow\mathcal{P}$ is harmonic. (See [this book](http://www.math.jhu.edu/~js/Math748/helein.harmonic.pdf) for a reference)


But, I can't find any references that reverse this relationship. Namely, if $f:\mathcal{M}\rightarrow\mathcal{N}$ is harmonic and $\phi:\mathcal{N}\rightarrow\mathcal{P}$ is (weakly) conformal, then is $\phi\circ f:\mathcal{M}\rightarrow\mathcal{P}$ harmonic?



Harmonic/conformal map composition between manifolds in either order?
Because it is not true. For example, suppose that $f: D^n \to D^n$ (the unit disk in $R^n$ with Euclidean metric) is harmonic -- i.e. the components are bounded harmonic functions in the ordinary sense satisfying $f\_1^2 + \ldots + f\_n^2 < 1$. Let $\phi: D^n \to D^n$ be the identity map, where the domain has the Euclidean and the range has the hyperbolic metric. Then $\phi \circ f$ is almost never harmonic.


