Canonical parameterizations of metric disks

Abstract

We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Generalizations and applications to the geometry of such surfaces are described.

Maximal metric surfaces and the Sobolev-to-Lipschitz property

“…Recall that a map is monotone if the preimage of every point is a connected set. We remark that in [35] the result is proven for M = D. The same proof however carries over to S 2 with minor modifications.…”

“…A celebrated result due to Bonk and Kleiner [3] states that an Ahlfors 2-regular metric sphere Z is quasisymetrically equivalent to the standard sphere if and only if it is linearly locally connected. Recently, it was shown in [35] that the quasisymmetric homeomorphism u Z : S 2 → Z may be chosen to be of minimal energy E 2 + (u Z ), and in this case is unique up to a conformal diffeomorphism of S 2 .…”

“…More generally, an Ahlfors 2-regular disc Z with finite boundary length admits a quasisymmetric parametrization by the standard disc D if and only if it is linearly locally connected. The parametrization may again be chosen to be of minimal Reshetynak energy and is then unique up to a conformal diffeomorphism, see [35]. A suitable variant of Theorem 1.2 also holds in this setting.…”

“…Indeed, P u is infinitesimally quasiconformal, Hölder continuous and its minimal weak upper gradient has global higher integrability; cf. [31][32][33]35]. Compared to the construction in [33], we trade off some regularity of u in exchange for the Sobolev-to-Lipschitz property on Z u .…”

“…The definition of general cell-like sets may be found in [33] but for our purposes it suffices to know that a compact subset K of S 2 is cell-like if and only if K and S 2 \ K are connected. Let us recall the following result from [35]. In the following we fix such energy minimzing quasisymmetric homeomorphism u ∈ N 1,2 (S 2 , Z ).…”