A couple of weeks ago, my student Yan Mary He presented a nice proof of Liouville’s theorem to me during our weekly meeting. The proof was the one from Benedetti-Petronio’s Lectures on Hyperbolic Geometry, which in my book gets lots of points for giving careful and complete details, and being self-contained and therefore accessible to beginning graduate students. Liouville’s Theorem is the fact that a conformal map between open subsets of Euclidean space of dimension at least 3 are Mobius transformations — i.e. they look locally like the restriction of a composition of Euclidean similarities and inversions on round spheres. This implies that the image of a piece of a plane or round sphere is a piece of a plane or round sphere, a highly rigid constraint. This sort of rigidity is in stark contrast to the case of conformal maps in dimension 2: any holomorphic (or antiholomorphic) map between open regions in the complex plane is a conformal map (and conversely). The proof given in Benedetti-Petronio is certainly clear and readable, and gives all the details; but Mary and I were a bit unsatisfied that it did not really provide any geometric insight into the meaning of the theorem. So the purpose of this blog post is to give a short sketch of a proof of Liouville’s theorem which is more geometric, and perhaps easier to remember.
Remember that a conformal map is one which infinitesimally takes round spheres to round spheres. That is, it is angle preserving, at least infinitesimally. In particular, it is smooth. So let’s think about a conformal map between open regions in Euclidean 3-space (for concreteness). The image of a flat plane P is a smooth surface f(P). Pick a point p in P and look at its image f(p). Infinitesimal round circles around p in P get taken to infinitesimal round circles around f(p) in f(P). And straight lines perpendicular to P get taken to smooth curves perpendicular to f(P). If you take a smooth surface S in Euclidean 3-space, and a small round circle in S, and push the circle off S in the perpendicular direction, some directions will be distorted more than others (typically): the infinitesimal circle gets distorted to an infinitesimal ellipse, whose major and minor axes are the directions of principal curvature on the surface S. But these ellipses are the conformal image of small round circles in the domain, and therefore should also be (almost) round. In other words: the principal curvatures at each point of f(P) should be equal. A point on a surface where the principal curvatures are equal is called an umbilical point, and a surface on which every point is umbilical is called totally umbilical.
It is a classical fact, proved by Meusnier in 1785, that an umbilical surface in Euclidean space is locally a piece of a plane or sphere. One way to see this is as follows. Let G denote the Gauss map, so that the condition of being umbilical at a point says exactly that dG is a multiple of the identity at that point (note: we are using here in the usual way the canonical identification between the tangent space to the surface and the tangent space to the round sphere at the image of the Gauss map at each point to think of dG as a map from the tangent space to itself). So if a surface is totally umbilical, there is some function f so that dG is equal to f times the identity at each point. Let’s denote by X a local chart on the surface giving rise to local coordinates u and v. So the definition of f says in this notation that and . But then
Since u and v are local coordinates, their tangent vectors and are independent, and therefore . This means that is (locally) constant. But this means that the surface osculates a sphere (or plane) of fixed curvature to first order at every point, and therefore (by developing this sphere along a path in the surface) the center of this osculating sphere is fixed and the surface agrees (locally) with the sphere (or plane). Incidentally, Gauss was only 8 years old in 1785, so whatever Meusnier’s proof was, he could not have mentioned the Gauss map by name. Does any reader know Meusnier’s argument?
Once we know that a conformal map takes subsets of planes and round spheres to subsets of planes and round spheres, we can intersect these planes and spheres with perpendicular planes and spheres to see that it takes straight segments and arcs of round circles to straight segments and arcs of round circles. From this it is easy to deduce Liouville’s theorem.
By the way, I strongly suspect that the connection between totally umbilical surfaces and conformal maps is classical and well-known, and for all I know this was how Liouville thought of his theorem in the first place.