Complex geodesics in convex domains and $\mathbb C$-convexity of semitube domains

TL;DR

This paper investigates the properties of complex geodesics in convex domains within complex spaces, providing geometric conditions for their characterization and exploring the $\,\mathbb{C}$-convexity of semitube domains.

Contribution

It introduces a geometric necessary condition for holomorphic maps to be complex geodesics in convex domains and derives explicit formulas for these geodesics.

Findings

Necessary geometric condition for complex geodesics

Explicit formulas for complex geodesics in convex domains

Discussion of $\,\mathbb{C}$-convexity of semitube domains

Abstract

In the paper the complex geodesics of a convex domain in $\mathbb C^n$ are studied. One of the main results of the paper provides certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in $\mathbb C^n$. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The $\mathbb C$-convexity of semitube domains is also discussed.