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 $\mathcal{F}$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal{F}$ 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 $\mathcal{F}$ contains a given point of $\overline{M}$ and $\mathcal{F}$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.

19 pages - final version, to appear in J. Geom. Anal