Abstract
We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau’s problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.
Highlights
1.1 Main resultSince Gromov stated his precompactness criterion in [7] the study of compact and precompact subsets of Gromov-Hausdorff space of metric spaces became a vivid field
We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle
We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds
Summary
Since Gromov stated his precompactness criterion in [7] the study of compact and precompact subsets of Gromov-Hausdorff space of metric spaces became a vivid field. In this article we investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and upper bound on the length of the boundary circle. Where D(L, C) denotes the closure of D(L, C) with respect to Gromov-Hausdorff distance It has been shown in [16] that a proper geodesic metric space X satisfies a (4π )−1-quadratic isoperimetric inequality iff it is a CAT(0)-space. In this light our Theorem 1.1 covers as a special case the compactness lemma in [19] which states that the class of CAT(0) disc retracts Z satisfying l(∂ Z ) ≤ L is compact.
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