Embeddedness of proper minimal submanifolds in homogeneous spaces

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ 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 $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \mathbb{R}^n$ bounded by $Γ_5$ is embedded. Let $Γ_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $Σ\subset \mathbb{H}^n$ bounded by $Γ_5$ is embedded. If $Γ_5$ is in a geodesic ball of radius $\fracπ{4}$ in $\mathbb{S}^n_+$, then $Σ\subset \mathbb{S}^n_+$ is also embedded. As a consequence, $Γ_5$ is an unknot in $\mathbb{R}^3$, $\mathbb{H}^3$ and $\mathbb{S}^3_+$. $({\rm ii})$ Let $Σ$ be an $m$-dimensional proper minimal submanifold in $\mathbb{H}^n$ with the ideal boundary $\partial_{\infty} Σ= Γ$ in the infinite sphere $\mathbb{S}^{n-1}=\partial_\infty \mathbb{H}^n$. If the M{ö}bius volume of $Γ$ $\widetilde{\vol}(Γ) < 2\vol(\mathbb{S}^{m-1})$, then $Σ$ is embedded. If $\widetilde{\vol}(Γ) = 2\vol(\mathbb{S}^{m-1})$, then $Σ$ is embedded unless it is a cone. $({\rm iii})$ Let $Σ$ be a proper minimal surface in $\hr$. If $Σ$ is vertically regular at infinity and has two ends, then $Σ$ is embedded.

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