# Pencils of circles and Liouville's theorem

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.

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That's a nice idea, but I'm afraid that these transformations won't preserve the measure used to define the average. –  Noam D. Elkies Feb 8 '13 at 20:02
That's right Noam, that's the snag. One would need to define the averages with respect to some Moebius-invariant "arc-length" that is non-zero on circles (and I don't know that there is any such thing) for which, moreover, the mean value theorem would hold. Still I feel it is very odd that the mean value theorem is formulated for a very specific class of elliptic pencils, when the space of harmonic functions is invariant under Moebius transformations. –  alvarezpaiva Feb 8 '13 at 20:49
Not an answer, just a related geometric argument that you might enjoy (or know already), by Edward Nelson, PAMS 1961: "Consider a bounded harmonic function on Euclidean space. Since it is harmonic, its value at any point is its average over any sphere, and hence over any ball, with the point as center. Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume."...(continued) –  Goldstern Feb 8 '13 at 23:43
(part 2 of Nelson's argument:) "... Since the function is bounded, the averages of it over the two balls are arbitrarily close, and so the function assumes the same value at any two points. Thus a bounded harmonic function on Euclidean space is a constant." jstor.org/stable/2034412 –  Goldstern Feb 8 '13 at 23:45
@Goldstern: it was Nelson's proof (+ Hadamard's three circle theorem) that prompted to think that maybe the conformal geometry in the space of circles was appearing implicitly in the study of harmonic functions. –  alvarezpaiva Feb 9 '13 at 10:34