Convexity properties and complete hyperbolicity of Lempert's elliptic tubes

TL;DR

This paper demonstrates that elliptic tubes over convex domains in real projective space are C-convex and exhibit complete Kobayashi-hyperbolicity, and explores their role in complexifying convex real projective manifolds.

Contribution

It establishes convexity and hyperbolicity properties of elliptic tubes and introduces a natural complexification method for convex real projective manifolds.

Findings

Elliptic tubes are C-convex.

Elliptic tubes are complete Kobayashi-hyperbolic.

A natural complexification of convex real projective manifolds is studied.

Abstract

We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.