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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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