Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics

TL;DR

This paper investigates the conditions under which holomorphic maps that preserve complex geodesics between Kobayashi hyperbolic manifolds are biholomorphic, focusing on domains like convex domains, the unit ball, and specific dimensions.

Contribution

It establishes slice rigidity properties for holomorphic maps preserving complex geodesics in certain classes of hyperbolic manifolds, including convex domains and the unit ball.

Findings

Holomorphic maps preserving geodesics are biholomorphisms under specific conditions.

Results extend to dimension 2 and the unit ball.

Conditions involve convexity, boundary points, and spanning properties.

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.