Is there any relation (maybe implicit) between the conformal geometry in the space of circles and spheres and the study of harmonic functions?

In the original question I was musing whether the following argument could be made into **rigorous** proof of Lioville's theorem (*any bounded harmonic function on the plane (or $\mathbb{R}^n$) is constant.*)?

Geometer "proof" (i.e. just as bad as a physicist's "proof"):

By the mean value theorem, the average of a harmonic function over any two concentric circles is the same. Notice that (1) a pencil of concentric circles is just an elliptic pencil of circles where one of the limit points is infinity, and that (2) all elliptic pencils are equivalent by some Moebius transformation. Since Moebius transformations take harmonic functions to harmonic functions, the mean value theorem actually says that the average of a harmonic function over any two circles in an elliptic pencil is the same. To see that the value of the function at any two points is the same, apply this remark to an elliptic pencil of circles having these two points as limit points.

Obviously this is **not** a proof : those Moebius transformations are moving infinity all over the place and some of the circles in the elliptic pencils are actually straight lines. However, I never used the fact that the harmonic function is bounded either ! Can one use the hypothesis that the function is bounded to make these ideas rigorous?

Disclaimer: this is just for fun.

Noam Elkies pointed out (see his comment) that there is the additional difficulty that the way we measure averages on circles depends also on the point at infinity and is not preserved by Moebius transformations.