Isometries for the Caratheodory Metric.pdf

a r X i v : 0 7 0 7 .2 3 2 9 v 2 [ m a t h .F A ] 1 9 S e p 2 0 0 7 ISOMETRIES FOR THE CARATHE?ODORY METRIC MARCO ABATE AND JEAN-PIERRE VIGUE? 1. Introduction The following problem has been studied by many authors. Let D1 and D2 be two bounded domains in complex Banach spaces and let f : D1 → D2 be a holomorphic map such that f ′(a) is a surjective isometry for the Carathe?odory infinitesimal metric at a point a of D1. The problem is to know whether f is an analytic isomorphism of D1 onto D2. For example, J.-P. Vigue? [8] proved this is the case when D1 and D2 are two bounded domains in C n and D1 is convex. Similar results have been obtained when D2 is convex using the Kobayashi infinitesimal metric (I. Graham [3] and L. Belkhchicha [1]). We have to remark that all these results are based on the theorem of L. Lempert ([5] et [6]; one can also consult M. Jarnicki and P. Pflug [4]) on the equality of Kobayashi and Carathe?odory metrics on a bounded convex domain in Cn. J.-P. Vigue? [9] proved the first results on this subject in the case of bounded domains in complex Banach spaces. Now, we can study the same problem dropping the hypothesis that f ′(a) is surjective. So, we only suppose that f ′(a) is an isometry for the Carathe?odory infinitesimal metric. Does this imply that f(D1) is a complex analytic closed submanifold of D2 and that f is an analytic isomorphism of D1 onto f(D1)? Some results have been obtained by J.-P. Vigue? [10] and P. Mazet [7] assuming that D1 and D2 are open unit balls in complex Banach spaces, that a = 0, and that the image of f ′(0) contains enough complex extremal points of the boundary of D2. Under these hypotheses they proved that f is linear equal to f ′(0). This result shows that f(D1) is an analytic submanifold of D2 and that f is an analytic isomorphism of D1 onto f(D1). Of course, if we do not suppose the existence of complex extremal points in the image of f ′(0), the map f has no reason to be linear. However, one can hope