Intrinsic structure of minimal discs in metric spaces

Abstract

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces. 7 2. Preliminaries 2.1. Basic notation. The following notation will be used throughout the paper. The Euclidean norm of a vector v ∈ R n is denoted by |v|.We denote the open unit disc in R 2 by D. A domain will always mean an open, bounded, connected subset of R 2 .Metric spaces appearing in this paper will be assumed complete. A metric space is called proper if its closed bounded subsets are compact. We will denote distances in a metric space X by d or d X . Let X = (X, d) be a metric space. The open ball in X of radius r and center x 0 ∈ X is denoted by B(x 0 , r) = B X (x 0 , r) = {x ∈ X : d(x 0 , x) < r}.

Isoperimetric characterization of upper curvature bounds

“…We refer to [LW16a], [LW16c] and Section 6 below for the notion of a solution of the Plateau problem and the associated intrinsic minimal disc. By definition of the intrinsic minimal disc Z, the solution of the Plateau problem u in Theorem 1.2 factorizes as u =ū•P for a surjective map P :D → Z and a 1-Lipschitz mapū : Z → X. Moreover,ū sends the boundary circle ∂Z of Z in an arclength preserving way onto Γ.…”

“…Applying some cutting and pasting tricks we reduce the final step to the question whether the length of the boundary curve "is controlled by the conformal factor". Using general structural results about the intrinsic minimal discs obtained in [LW16c], the final step reduces to the following: Theorem 1.3. Let Z be a geodesic metric space homeomorphic to the closed discD.…”

Spaces with almost Euclidean Dehn function

“…Roughly speaking, this candidate comes as the ultralimit of a sequence of area minimizers in X whose boundaries form a sequence of Lipschitz curves in X approximating Γ in X ω . This step relies on the results in [24] and [26] on the existence, regularity and equi-compactness of area minimizers in proper metric spaces. In a second step we show that the so found candidate is indeed an area minimizer.…”

“…By [26,Lemma 4.8], we have FillArea X ω (c) ≤ Area X ω (w ′ ) for all w ′ ∈ Λ(Γ, X ω ), which shows that u minimizes area among all elements in Λ(Γ, X ω ). This completes the proof of Theorem 6.3.…”

Metric-minimizing surfaces revisited

“…This lemma is nearly identical to [16,Corollary 7.12]; it could be considered a disc version of Moore's quotient theorem [19], [7] which states that if a continuous map f from the sphere S 2 to a Hausdorff space X has acyclic fibers, then f can be approximated by a homeomorphism; in particular X is homeomorphic to S 2 .…”