## Canonical parameterizations of metric disks.(English)Zbl 1451.30118

This paper studies parameterizations of metric surfaces. The authors provide a new proof of a result by Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected Ahlfors $$2$$-regular metric spaces. More precisely, they show that an Ahlfors $$2$$-regular metric space $$X$$ that is homeomorphic to the standard $$2$$-sphere admits a quasisymmetric homeomorphism from $$S^2$$ to $$X$$ if and only if $$X$$ is linearly locally connected, [M. Bonk and B. Kleiner, Invent. Math. 150, No. 1, 127–183 (2002; Zbl 1037.53023)].
The approach taken in the paper under review is generalizing the existence proof of conformal parameterizations of smooth surfaces using energy minimizers. In the first result, the authors assume that $$X$$ is a geodesic metric space that is homeomorphic to the closure of the unit disk and its boundary circle is of finite length, is Ahlfors $$2$$-regular, and is linearly locally connected. Under these assumptions, there exists a homeomorphism for the closed unit disk to $$X$$ that has minimal energy. Moreover such a homeomorphism is quasisymmetric and it is unique up to a conformal diffeomorphism of the closed unit disk. That is the content of their Theorem 1.1.
Theorem 1.2 gives sufficient conditions for an energy minimizer to be a uniform limit of a sequence of homeomorphisms.
One result that goes into the proof of Theorem 1.1 is their Theorem 1.4 and as follows. Assume $$X$$ is a geodesic metric space homeomorphic to the closed unit disk, the $$2$$-sphere, or the plane. Further require that there is a constant $$C$$ such that every Jordan curve in $$X$$ bounds a Jordan domain $$U\subset X$$ with $\mathcal{H}^2(U)\leq C\ell(\partial U)^2.$ Then $$X$$ admits a quadratic isoperimetric inequality with constant $$C$$.
As corollary of Theorems 1.2 and 1.4, the authors obtain a characterization for CAT(0) spaces in terms of isoperimetric inequalities. Under the same conditions as in Theorem 1.4, the space has the Hölder extension property for each Hölder exponent.
After three introductory sections introducing the results, general preliminaries and metric-space-valued Sobolev maps, the authors start with the proofs. Section 4 investigates topological properties of minimizers leading to the proof of Theorem 1.2. This is followed by the study of isoperimetric inequalities and the proof of Theorem 1.4. The final section details results about parameterizations. Finally, the appendix proves a result concerning deducing quasisymmetry from a modulus inequality.

### MSC:

 30L10 Quasiconformal mappings in metric spaces 58E20 Harmonic maps, etc. 49Q05 Minimal surfaces and optimization 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations

Zbl 1037.53023
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