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.


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