Slice Rigidity Property of Holomorphic Maps Kobayashi-Isometrically Preserving Complex Geodesics

Abstract

In this paper, we study the following “slice rigidity property”: given two Kobayashi complete hyperbolic manifolds M, N and a collection of complex geodesics \({\mathscr {F}}\) of M, when is it true that every holomorphic map \(F:M\rightarrow N\) which maps isometrically every complex geodesic of \({\mathscr {F}}\) onto a complex geodesic of N is a biholomorphism? Among other things, we prove that this is the case if M, N are smooth bounded strictly (linearly) convex domains, every element of \({\mathscr {F}}\) contains a given point of \({\overline{M}}\) and \({\mathscr {F}}\) spans all of M. More general results are provided in dimension 2 and for the unit ball.