## Embeddedness of proper minimal submanifolds in homogeneous spaces.(English)Zbl 1282.53055

Let $$\Gamma_{2m+1}$$ be a piecewise geodesic Jordan curve with $$2m+1$$ vertices in $$\mathbb R^n$$, $$\mathbb H^n$$ or $$\mathbb S^n$$, where $$m\geq 2$$ is an integer. The author proves three embeddedness results.
(i) The total curvature of $$\Gamma_{2m+1}\subset\mathbb R^n$$ is $$<2m\pi$$. 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. For $$\Gamma_5\subset\mathbb H^n$$, any minimal surface $$\Sigma\subset\mathbb H^n$$ bounded by $$\Gamma_5$$ is embedded. If $$\Gamma_5$$ is in a geodesic ball of radius $$\pi/4$$ in $$\mathbb S^n_+$$, then $$\Sigma\subset S^n_+$$ is also embedded. As a consequence, $$\Gamma_5$$ is an unknot in $$\mathbb R^3$$, $$\mathbb H^3$$, and $$S^3_+$$.
(ii) Let $$\Sigma$$ be an $$m$$-dimensional proper minimal submanifold in $$H^n$$ with the ideal boundary $$\partial_\infty\Sigma=\Gamma$$ in the infinite sphere $$\mathbb{S}^{n-1}=\partial_\infty H^n$$. If the Möbius volume of $$\Gamma$$ obeys $$\Gamma\widetilde{\text{Vol}}(\Gamma)< 2{\text{Vol}}(\mathbb{S}^{m-1})$$, then $$\Sigma$$ is embedded, and if the equality $$\widetilde{\text{Vol}}(\Gamma)=2{\text{Vol}}(\mathbb{S}^{m-1})$$ holds, then $$\Sigma$$ is embedded unless it is a cone.
(iii) If a proper minimal surface $$\Sigma$$ in $$H^2\times R$$ is vertically regular at infinity and has two ends, then $$\Sigma$$ is embedded.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C30 Differential geometry of homogeneous manifolds
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