| Title: | Domains without parabolic minimal submanifolds and weakly hyperbolic domains |
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| Authors: | ID Forstnerič, Franc (Author) |
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| Files: |  PDF - Presentation file, download (242,50 KB)
MD5: 968AF422A5EFB57D0AF9C573CD9E8476
 URL - Source URL, visit https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12894
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | minimal surfaces, m-plurisubharmonic functions, hyperbolic domain |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.12.2023 |
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| Year of publishing: | 2023 |
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| Number of pages: | str. 2778-2792 |
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| Numbering: | Vol. 55, iss. 6 |
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| PID: | 20.500.12556/DiRROS-18651  |
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| UDC: | 517.5 |
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| ISSN on article: | 0024-6093 |
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| DOI: | 10.1112/blms.12894 |
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| COBISS.SI-ID: | 161694467 |
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| Note: |
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| Publication date in DiRROS: | 10.04.2024 |
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| Views: | 1454 |
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| Downloads: | 924 |
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