Normal family theory and Gauss curvature estimate of minimal surfaces in \(\mathbb{R}^m\)
A fascinating result in complex function theory is Bloch's heuristic principle. According to this principle, a family of meromorphic functions which have a property \(\mathcal{P}\) in common on a domain \(U\subset\mathbb{C}\) should be normal in \(U\) if the property \(\mathcal{P}\) forces a meromorphic function defined on the whole complex plane \(\mathbb{C}\) to be constant. In the theory of minimal surfaces, a similar phenomenon holds. Every time while there was a Bernstein-type result concerning complete minimal surfaces satisfying a certain condition, there was a corresponding Gauss curvature estimate for the minimal surfaces with boundary that satisfy the same condition, i.e., \[ |K(p)|d(p)^2\leq C, \] where \(C\) is a positive constant, \(K\) the Gauss curvature and \(d(p)\) the geodesic distance from the point \(p\) to the boundary. Inspired by Bloch's principle, the authors derive various interesting results for minimal surfaces in \(\mathbb{R}^m\) whose Gauss maps satisfy some property \(\mathcal{P}\). Before stating the main result let us introduce some notation and terminology. Let \(\Sigma\) be a connected Riemann surface, \(\mathbb{P}^{m-1}\) be the complex projective space of dimension \(m-1\), \(D\) the open unit disc in \(\mathbb{C}\), and \(\mathcal{H}(\Sigma)\) the set of holomorphic maps from \(\Sigma\) to \(\mathbb{P}^{m-1}\). If \(\mathcal{P}\) is a property for holomorphic maps, we denote by \(\mathcal{P}(\Sigma)\) the set given by \[ \mathcal{P}(\Sigma) = \{g \in \mathcal{H}(\Sigma): g\,\, \text{satisfies the property}\,\, \mathcal{P}\}. \] We say that \(\mathcal{P}\) is a compact property if the following conditions are fulfilled: i) For any pair of Riemann surfaces \(\Sigma_1\), \(\Sigma_2\) and for any holomorphic map \(\phi : \Sigma_1 \to \Sigma_2\) without branch points, one has \(g\in\mathcal{P}(\Sigma_2)\Rightarrow g\circ\phi\in\mathcal{P}(\Sigma_1)\). ii) Let \(\Sigma\) be any Riemann surface and \(g \in \mathcal{H}(\Sigma)\). For any relatively compact domain \(\Omega\) of \(\Sigma\) one has \(g|_{\Omega}\in\mathcal{P}(\Omega)\Rightarrow g\in\mathcal{P}(\Sigma).\) iii) For any Riemann surface \(\Sigma\), \(\mathcal{P}(\Sigma)\) is compact in \(\mathcal{H}(\Sigma)\). One of the main results of the paper is the theorem below. \textbf{Theorem:} Let \(\mathcal{P}\) be a compact property for holomorphic maps from Riemann surfaces into \(\mathbb{P}^{m-1}\). Then the following alternative holds: \(a)\) \(\mathcal{P}\) satisfies a Gauss curvature estimate, or \(b)\) there is a conformal complete minimal immersion \(x : D \to \mathbb{R}^m\) whose Gauss map lies in \(\mathcal{P}(D)\) and whose Gauss curvature \(K\) satisfies \(|K(0)| = 1\) and \(|K| \leq 4\) in \(D\). Using the above theorem, the authors recover and simplify known results by \textit{R. Osserman} and \textit{M. Ru} [J. Differ. Geom. 46, No. 3, 578--593 (1997; Zbl 0918.53003)] about value distribution properties of the Gauss map of minimal surfaces in \(\mathbb{R}^m\).
scientific article; zbMATH DE number 6603545