Hyperbolic domains in real Euclidean spaces

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.

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The first-named author is supported by the research program P1-0291 and grants J1-3005, N1-0137, and N1-0237 from ARIS, Republic of Slovenia.

The second-named author is supported by the European Union (ERC Advanced grant HPDR, 101053085) and the research program P1-0291 and grants J1-3005, and N1-0237 from ARIS, Republic of Slovenia.

Received 22 November 2021

Accepted 21 January 2022

Published 30 January 2024