Abstract

We give a new approach to the study of relations between the Gauss map and compactness properties for families of minimal surfaces in the Euclidean three space. In particular, we give a simple and unified proof of the curvature estimates for stable minimal surfaces and for minimal surfaces whose Gauss map image omits five points. The Gauss map of a minimal surface in the Euclidean space R 3 is a conformal map. This fact has deep consequences in the behavior of these surfaces and has allowed a massive presence of complex variable techniques in the classical theory of minimal surfaces. In 1959 Osserman 19 started a systematic study of this map, showing that the Gauss map of a non-flat complete minimal surface must be dense in the unit sphere S 2 .H e also proved 20 the following curvature estimate, which clearly implies the result above: There exists a positive constant C> 0 such that, for all minimal surfaces ψ :Σ → R 3 whose Gauss map omits a fixed geodesic disc in S 2 and for all point p ∈ Σ, we have

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