Embeddedness of proper minimal submanifolds in homogeneous spaces

Abstract

We prove the three embeddedness results as follows. $(\mathrm{i})$ Let ${\Gamma }_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in ${\mathbb{R}}^n$, where $m$ is an integer $\ge 2$. Then the total curvature of ${\Gamma }_{2m+1}<2m\pi$. In particular, the total curvature of ${\Gamma }_5<4\pi$ 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 ${\mathbb{S}}_{+}^3$. $(\mathrm{ii})$ 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{ö}bius volume of $\Gamma$ $\widetilde{\vol}(Γ) < 2\vol(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded. If $\widetilde{\vol}(Γ) = 2\vol(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded unless it is a cone. $(\mathrm{iii})$ Let $\Sigma$ be a proper minimal surface in $\hr$. If $\Sigma$ is vertically regular at infinity and has two ends, then $\Sigma$ is embedded.

20 pages, 2 figures