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- W1551097706 abstract "We prove the three embeddedness results as follows. $({rm i})$ Let $Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $mathbb{R}^n$, where $m$ is an integer $geq2$. Then the total curvature of $Gamma_{2m+1}<2mpi$. In particular, the total curvature of $Gamma_5<4pi$ and thus any minimal surface $Sigma subset mathbb{R}^n$ bounded by $Gamma_5$ is embedded. Let $Gamma_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $mathbb{H}^n$. Then any minimal surface $Sigma subset mathbb{H}^n$ bounded by $Gamma_5$ is embedded. If $Gamma_5$ is in a geodesic ball of radius $frac{pi}{4}$ in $mathbb{S}^n_+$, then $Sigma subset mathbb{S}^n_+$ is also embedded. As a consequence, $Gamma_5$ is an unknot in $mathbb{R}^3$, $mathbb{H}^3$ and $mathbb{S}^3_+$. $({rm ii})$ Let $Sigma$ be an $m$-dimensional proper minimal submanifold in $mathbb{H}^n$ with the ideal boundary $partial_{infty} Sigma = Gamma$ in the infinite sphere $mathbb{S}^{n-1}=partial_infty mathbb{H}^n$. If the M{o}bius volume of $Gamma$ $widetilde{vol}(Gamma) < 2vol(mathbb{S}^{m-1})$, then $Sigma$ is embedded. If $widetilde{vol}(Gamma) = 2vol(mathbb{S}^{m-1})$, then $Sigma$ is embedded unless it is a cone. $({rm iii})$ Let $Sigma$ be a proper minimal surface in $hr$. If $Sigma$ is vertically regular at infinity and has two ends, then $Sigma$ is embedded." @default.
- W1551097706 created "2016-06-24" @default.
- W1551097706 creator A5065022979 @default.
- W1551097706 date "2010-11-18" @default.
- W1551097706 modified "2023-10-14" @default.
- W1551097706 title "Embeddedness of proper minimal submanifolds in homogeneous spaces" @default.
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- W1551097706 doi "https://doi.org/10.48550/arxiv.1011.4140" @default.
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