Elliptic Equations in Conformal Geometry

A conformal transformation is a diffeomorphism which preserves angles; the differential at each point is the composition of a rotation and a dilation. In its original sense, conformal geometry is the study of those geometric properties preserved under transformations of this type. This subject is deeply intertwined with complex analysis for the simple reason that any holomorphic function f(z) of one complex variable is conformal (away from points where f ′(z) = 0), and conversely, any conformal transformation from one neighbourhood of the plane to another is either holomorphic or antiholomorphic. This provides a rich supply of local conformal transformations in two dimensions. More globally, the (orientation preserving) conformal diffeomorphisms of the Riemann sphere S are its holomorphic automorphisms, and these in turn constitute the noncompact group of linear fractional transformations. By contrast, the group of conformal diffeomorphisms of any other compact Riemann surface is always compact (and finite when the genus is greater than 1). Implicit here is the notion that a Riemann surface is a smooth two-dimensional surface together with a conformal structure, i.e. a fixed way to measure angles on each tangent space. There is a nice finite dimensional structure on the set of all inequivalent conformal structures on a fixed compact surface; this is the starting point of Teichmüller theory. All of this is in accord with the fact that the conformal property is an elliptic equation in two dimensions, so there are many solutions locally, but global existence is constrained and solutions lie in finite dimensional families. In higher dimensions, the equation that a map be conformal is formally overdetermined, so one expects very few such transformations, if any. Generalizations of the linear fractional transformations always do exist: for any n, the conformal group O(n+ 1, 1) – also called the Möbius group – consists of all mappings on R which are compositions of translations, rotations, dilations and inversions about spheres. Any Möbius transformation extends naturally both to a global conformal diffeomorphism of S, and also to its interior, the ball, as an isometry for the hyperbolic Poincaré metric on B. This last statement is the basis for the far-reaching correspondence between hyperbolic manifolds and conformal group actions on the sphere, which has many ramifications in geometry, dynamics and number theory. However, these are the only conformal mappings of the sphere; a theorem due to Liouville states that any locally defined conformal diffeomorphism in R with n > 2 is the restriction of one of these Möbius transformations. (In particular, except