Domains without parabolic minimal submanifolds and weakly hyperbolic domains
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910678573383680 |
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| author | Forstneric, Franc |
| author_facet | Forstneric, Franc |
| contents | We show that if $Ω$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $bΩ$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $Ω$ does not contain any immersed parabolic minimal submanifold of dimension $\ge m$. In particular, if $M$ is a properly embedded nonflat 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. We also introduce the class of weakly hyperbolic domains $Ω$ in $\mathbb R^n$ characterised by the property that every conformal harmonic map $\mathbb C\toΩ$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_04689 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Domains without parabolic minimal submanifolds and weakly hyperbolic domains Forstneric, Franc Differential Geometry Complex Variables Primary 53A10. Secondary 32Q45 We show that if $Ω$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $bΩ$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $Ω$ does not contain any immersed parabolic minimal submanifold of dimension $\ge m$. In particular, if $M$ is a properly embedded nonflat 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. We also introduce the class of weakly hyperbolic domains $Ω$ in $\mathbb R^n$ characterised by the property that every conformal harmonic map $\mathbb C\toΩ$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces. |
| title | Domains without parabolic minimal submanifolds and weakly hyperbolic domains |
| topic | Differential Geometry Complex Variables Primary 53A10. Secondary 32Q45 |
| url | https://arxiv.org/abs/2207.04689 |