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An estimate for the Gauss curvature of minimal surfaces in \(\mathbf R^m\) whose Gauss map omits a set of hyperplanes. (English) Zbl 0918.53003

Let \(x : M \to \mathbb{R}^m\) be a minimal surface immersed in \( \mathbb{R}^m\) and let \(g : M \to Q_{m-2}(\mathbb{C}) \subset P^{m-1}(\mathbb{C})\) be the generalized Gauss map of \(M\), where \(Q_{m-2}(\mathbb{C})\) denotes the quadric in \(P^{m-1}(\mathbb{C})\). For \(m=3\), \(Q _1 (\mathbb{C})\) is canonically identified with the unit sphere \(S^2\) in \(\mathbb{R}^3\) and \(g\) with the classical Gauss map of \(M\). In 1988, H. Fujimoto [Sugaku Expo. 6, 1-13 (1993); translation from Sugaku 40, 312-321 (1988; Zbl 0629.53011)] showed that if the Gauss map of a minimal surface in \(\mathbb{R}^3\) omits five distinct values in \(S^2\), then \[ | K(p)|^{1/2} d(p) \leq c, \tag{\(*\)} \] where \(K(p), d(p)\) denote the Gaussian curvature of \(M\) at \(p\) and the distance from \(p\) to the boundary of \(M\), respectively, and \(c\) is a constant, depending on the set of omitted values. In 1991, M. Ru [J. Differ. Geom 34, 411-423 (1991; Zbl 0723.53005)] proved that if the generalized Gauss map of a complete minimal surface in \( \mathbb{R}^m\) omits more than \(m(m+1)/2\) hyperplanes in \(P^{m-1}(\mathbb{C})\), located in general position, then \(g\) is constant and the minimal surface must be a plane.
In the present paper, the authors prove that if the Gauss map of a minimal surface immersed in \( \mathbb{R}^m\) omits more than \(m(m+1)/2\) hyperplanes in \(P^{m-1}(\mathbb{C})\), located in general position, then the inequality \((*)\) holds for a constant \(c\), depending on the set of omitted hyperplanes. For the proof of this result, the authors prove an interesting lemma, which may be viewed as a kind of Ahlfors form of the Schwarz-Pick lemma.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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