On the compactness of isometry groups in complex analysis

Abstract

We prove that the group of continuous isometries for the Kobayashi or Caratheodory metrics of a strongly convex domain in ℂⁿ is compact unless the domain is biholomorphic to the ball. A key ingredient, proved using differential geometric ideas, is that a continuous isometry between a strongly convex domain and the ball has to be biholomorphic or anti-biholomorphic. Combining this with a metric version of Pinchuk's rescaling technique gives the main result.