Quantitative quasisymmetric uniformization of compact surfaces

Abstract

Abstract. Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere, in a quantitative way. We extend this result to arbitrary metric compact orientable surfaces.

Uniformization of metric surfaces using isothermal coordinates

“…Theorem 1.2 of [GW18] proves that if (Y, d Y ) is as in the statement of Theorem 1.4, orientable and not homeomorphic to S 2 , there exists a complete Riemannian surface Z of constant curvature and an η ′ -quasisymmetric homeomorphism φ : Z → (Y, d Y ) with η ′ depending only on the data of (Y, d Y ). Using Theorem 1.3, our isothermal coordinates, and a modified version of their proof, we prove that the uniformization map is η-quasisymmetric with η depending only on the data of (Y, d Y ).…”

“…Proof of Claim (1): If Ω is the hyperbolic disk or the Euclidean plane, the existence of β ′ and η follows from Propositions 5 and 7 of [GW18] (which are stated for the case when ψ is orientable. However, the non-orientable case follows from the orientable one by applying the conjugate map z → z in the Euclidean unit disk D).…”

“…This can be done without loss of generality since the properties we study are left unchanged by rescaling. The diameter normalization is needed for the results we use from [GW18]. We formulate the following corollary of [GW18, Theorem 9] and Lemma 6.5.…”

Canonical parametrizations of metric surfaces of higher topology

“…The theorem of Bonk-Kleiner has been extended for example in [19], [16] and [20]. In particular, there are analogous quasisymmetric uniformization results for Ahlfors 2-regular (orientable) closed surfaces, see [8] and [11]. Theorem 1.1 is a strengthening of these results, since it states the existence of canonical quasisymmetric homeomorphisms, even for such surfaces with boundary.…”

A metric sphere not a quasisphere but for which every weak tangent is Euclidean

“…An alternative proof of Theorem 1.4 can be given using ideas in [GW18]. Roughly speaking, Theorem 1.4 says local geometry properties promote to global property.…”