Riv.Mat.Univ.Parma (4) 17 (1991)


The space of spheres and conformal geometry

In memory of G. I. Katz

Pages: 111-130
Received: 18 September 1990

Abstract We show that: (1) the complement of the limit set of a Kleinian group acting in Sn carries an invariant Finsler metric; (2) every conformal map of a compact domain in Sn in itself is a contraction of some Finsler metric given by an explicit formula; (3) for a length-parametrized smooth curve γ(t) in 2 the integral is a conformal invariant (K is the curvature), which is an "integrated" version of a conformal invariance of the Virasoro cocycle and Schwartzian; (4) every non-convex near infinity hypersurf in half-space of n should yield strong curvature decaying conditions; (5) natural representation of O(n + 1, 1) in n+1,1 is "glued" to some isometrical Banach representation (unitar when n=2), which is a sort of quantization of "gluing" the conformal action on Sn to isometrical action in the hyperbolic space Hn+1.