Compactification of minimal submanifolds of hyperbolic space
The author studies the geometry of complete minimal immersions \(\varphi: M^ n \hookrightarrow H^ n\) of an \(m\)-dimensional manifold \(M\) into hyperbolic space \(H^ n\) under the assumption of finite total ``extrinsec'' curvature \(\int_ M | A|^ m<\infty\), \(| A|\) denoting the norm of the second fundamental form. The investigations complement results of Osserman, Chern-Osserman and Anderson on complete finite total curvature minimal submanifolds in Euclidean space. We state the two main results of the paper from the author's introduction: Theorem A. Let \(\varphi: M^ m \hookrightarrow \mathbb{H}^ n\) be a complete minimal immersion of a connected \(m\)-dimensional manifold \(M\). Suppose that \(\int_ M |{\mathcal A}|^ m <\infty\). Then \(M\) is properly immersed and is diffeomorphic to the interior of a compact manifold \(\overline{M}\) with boundary. Furthermore \(\varphi\) extends to a continuous map \(\overline{\varphi}: \overline{M} \hookrightarrow \overline {\mathbb{H}}^ n\), \(\overline {\mathbb{H}}^ n\) the compactified of \(\mathbb{H}^ n\). Theorem B. Let \(M^ 2 \hookrightarrow \mathbb{H}^ n\) be a complete connected minimal surface with \(\int_ M |{\mathcal A}|^ 2<\infty\). Then \(M\) is conformally equivalent to a compact surface \(\overline{M}\) with a finite number of disks removed and the index of the operator \({\mathbf L}= -\Delta+2- |{\mathcal A}|^ 2\) is finite. Furthermore the asymptotic boundary \(\partial_ \infty M\) is a Lipschitz curve. Sobolev inequalities and Simons' equation for the Laplacian of \(A\) on \(M\) are used to show that \(| A|(p)\to 0\) as \(\text{dist}_ M (p_ 0,p)\to\infty\) for any fixed point \(p_ 0\in M\). Further investigations of the induced distance function lead to the analytical and topological results of both theorems. Additionally finiteness of the index of the operator \(L=- \Delta+2- | A|^ 2\) is shown.
scientific article; zbMATH DE number 408734