Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 6

Special Issue in honor of Professor Blaine Lawson’s 80th birthday

Guest Editors: Shiu-Yuen Cheng, Paulo Lima-Filho, and Stephen Shing-Toung Yau

Hyperbolic domains in real Euclidean spaces

Pages: 2689 – 2735

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n6.a4

Authors

Barbara Drinovec Drnovšek (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia ; and Institute of Mathematics, Physics & Mechanics, Ljubljana, Slovenia)

Franc Forstnerič (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia ; and Institute of Mathematics, Physics & Mechanics, Ljubljana, Slovenia)

Abstract

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb{R}^n$, $n \geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine $2$-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.

Keywords

minimal surface, minimal metric, hyperbolic domain

2010 Mathematics Subject Classification

Primary 53A10. Secondary 30C80, 31A05, 32Q45.

Received 22 November 2021

Accepted 21 January 2022

Published 30 January 2024