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The local picture theorem on the scale of topology. (English) Zbl 1396.53088

From the introduction: This paper is devoted to an analysis of the extrinsic geometry of any embedded minimal surface \(M\) in small intrinsic balls in a homogeneously regular Riemannian three-manifold, such that the injectivity radius function of \(M\) is sufficiently small in terms of the ambient geometry of the balls. We carry out this analysis by blowing-up such an \(M\) at a sequence of points with shape almost-minimal injectivity radius, which produces a new sequence of minimal surfaces, a subsequence of which has a natural limit object being either a properly embedded minimal surface in \(\mathbb R^3\), a shape minimal parking garage structure on \(\mathbb R^3\) or possibly, a particular case of a singular minimal lamination of \(\mathbb R^3\) with restricted geometry.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
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