Domains without parabolic minimal submanifolds and weakly hyperbolic domains

Forstnerič, Franc (author)

We show that if $\Omega$ is an $m$-convex domain in $\mathbb{R}^n$ for some $2 \le m < n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ does not contain any immersed parabolic minimal submanifolds of dimension $\ge m$. In particular, if $M$ is a properly embedded non-flat minimal hypersurface in $\mathbb{R}^n$ with a tubular neighbourhood of positive radius, then every immersed parabolic hypersurface in $\mathbb{R}^n$ intersects $M$. In dimension $n=3$, this holds if $M$ has bounded Gaussian curvature function. We also introduce the class of weakly hyperbolic domains $\Omega$ in $\mathbb{R}^n$, characterised by the property that every conformal harmonic map $\mathbb{C} \to \Omega$ is constant, and we elucidate their relationship with hyperbolic domains, and domains without parabolic minimal surfaces.

• Language: English
• Keywords: minimal surfaces, $m$-plurisubharmonic functions, hyperbolic domain
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: Str. 2778-2792
• Numbering: Vol. 55, iss. 6
• PID: 20.500.12556/RUL-152922
• UDC: 517.5
• ISSN on article: 0024-6093
• DOI: 10.1112/blms.12894
• COBISS.SI-ID: 161694467
• Publication date in RUL: 12.12.2023

Record is a part of a journal

• Title: Bulletin of the London Mathematical Society
• Shortened title: Bull. Lond. Math. Soc.
• Publisher: Wiley, London Mathematical Society
• ISSN: 0024-6093
• COBISS.SI-ID: 25154560

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: minimalne ploskve, $m$-plurisubharmonične funkcije, hiperbolična domena