Totally geodesic discs in strongly convex domains

TL;DR

This paper proves that Kobayashi isometries between strongly convex domains in complex Euclidean spaces are necessarily holomorphic or anti-holomorphic, revealing a rigidity property of such isometries.

Contribution

It establishes a rigidity result showing that Kobayashi isometries in strongly convex domains must be holomorphic or anti-holomorphic, extending understanding of complex geometric mappings.

Findings

Kobayashi isometries are holomorphic or anti-holomorphic

Strongly convex domains exhibit rigidity in their isometries

Isometries preserve complex structure in these domains

Abstract

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $\Omega_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $\phi: (\Omega_1, d^K_{\Omega_1}) \rightarrow (\Omega_2, d^K_{\Omega_2})$ is an isometry, i.e. $ d^K_\Omega_{n_2}(f(\zeta),f(\eta)) = d^K_{n_1} (\zeta,\eta)$ for all $\zeta,\eta \in \Omega_1,$ then $\phi$ is either holomorphic or anti-holomorphic.