## 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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