This is a [cross-post](https://math.stackexchange.com/questions/2239890/conformal-harmonic-maps-in-high-dimensions-are-scaled-isometries) from MSE (where I got no answer).


It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.


I discovered lately that in dimension $d>2$, conformal harmonic maps must be scaled isometries (The conformal factor is constant).


I am quite sure this should be already known, but couldn't find a reference.
Any help?



Conformal harmonic maps in high dimensions are scaled isometries
This result is well-known in the theory of *harmonic morphisms*, about which, there is an extensive literature. It is a quite general fact (not depending on the conformally flat case of Euclidean space), implying that, when $n>2$, any conformal map $f:(M^n,g)\to (N^n,h)$ between (connected) Riemannian manifolds of dimension $n$ that is harmonic is necessarily a *homothety* i.e., $f^\*(h) = r^2 g$ for some constant $r\ge0$.


You can find proofs in introductory papers and books on harmonic morphisms. I'm traveling, so I don't have time to check the best references, but, for example, try B. Fuglede, *Harmonic morphisms between Riemannian manifolds*, Ann. Inst. Fourier 28 (1978), 107-144, which probably has a proof written out there.


You can also find a proof in the introductory sections of my paper, *Harmonic morphisms with fibers of dimension one* (Communications in Analysis and Geometry, Volume 8 (2000), 219–265, but you'll have to sort out the relevant bits of the discussion in order to trace through a proof.


