Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

中文翻译：

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最小超曲面的双曲展开

摘要 我们通过将这个主题与拟共形几何联系起来，从一个新的角度研究了面积最小化超曲面的内在几何。也就是说，对于任何这样的超曲面 H，我们定义并构建了一个所谓的 S 结构。这个新的和自然的概念揭示了 H 及其奇点集 Ʃ 的一些意想不到的几何和解析特性。此外，它可以用来证明 H\Ʃ 双曲展开的存在。这些是 H\Ʃ 的典型共形变形到有界几何的完整 Gromov 双曲空间，其中 Gromov 边界同胚于 Ʃ。这些新概念和结果自然会扩展到更大的几乎最小化类。