Quasiconformal parametrization of metric surfaces with small dilatation
aa r X i v : . [ m a t h . M G ] M a y QUASICONFORMAL PARAMETRIZATION OF METRICSURFACES WITH SMALL DILATATION
MATTHEW ROMNEY
Abstract.
We verify a conjecture of Rajala: if (
X, d ) is a metric surface oflocally finite Hausdorff 2-measure admitting some (geometrically) quasicon-formal parametrization by a simply connected domain Ω ⊂ R , then thereexists a quasiconformal mapping f : X → Ω satisfying the modulus inequal-ity 2 π − Mod Γ ≤ Mod f Γ ≤ π − Mod Γ for all curve families Γ in X . Thisinequality is the best possible. Our proof is based on an inequality for thearea of a planar convex body under a linear transformation which attains itsBanach-Mazur distance to the Euclidean unit ball. Introduction
A growing body of recent literature has studied the following quasiconformaluniformization problem : for a metric space (
X, d ) homeomorphic to a domain Ω in R n or S n , under what conditions does there exist a quasiconformal (or quasisym-metric) parametrization of X by Ω? This question originated largely in the workof Semmes; see especially [7] and [3, Qu. 3-7]. A landmark paper of Bonk andKleiner [2] gives a complete description of those spaces X admitting a quasisym-metric parametrization by S under the assumption that X is Ahlfors 2-regular; anecessary and sufficient condition is that X be linearly locally contractible.A similar theorem for geometrically quasiconformal parametrizations was re-cently proven by Rajala [6] in the setting of metric surfaces homeomorphic to R or S with locally finite Hausdorff 2-measure. A new condition called reciprocal-ity is introduced which is necessary and sufficient for the existence of the desiredquasiconformal parametrization. The original Bonk–Kleiner theorem can then beobtained as a corollary. We refer the reader to the introduction of Rajala’s paperfor additional background and references. See also Lytchak and Wenger [5] forother new results on quasiconformal parametrizations in somewhat the same spirit.We recall now the relevant definitions. Let ( X, d, µ ) be a metric measure space.Given a family Γ of curves in X , the p -modulus of Γ isMod p Γ := inf ρ Z X ρ p dµ, the infimum taken over all Borel functions ρ : X → [0 , ∞ ] such that R γ ρ ds ≥ γ ∈ Γ. A homeomorphism f : ( X, d, µ ) → ( Y, d ′ , ν ) is K -geometrically quasiconformal with exponent p if K − Mod p Γ ≤ Mod p f (Γ) ≤ K Mod p Γfor all curve families Γ in X . The smallest value K O such that Mod p Γ ≤ K O Mod p f (Γ)for all curve families Γ in X is called the outer dilatation of f . Similarly, the small-est value K I such that Mod p f (Γ) ≤ K I Mod p Γ for all curve families Γ in X is the inner dilatation . Mathematics Subject Classification.
Key words and phrases.
Quasiconformal mapping, conformal modulus, convex body. If p is understood, we say simply that f is K -quasiconformal or quasiconformal .In this note, we always take p = 2 and we write Mod Γ in place of Mod Γ. We willassume that a metric space (
X, d ) is equipped with the Hausdorff 2-measure.The same paper of Rajala also examines a related question: if such a quasi-conformal parametrization exists, can one find a quasiconformal mapping whichimproves the dilatation constants K O and K I to within some universal constants?If so, what is the best result of this type? Rajala obtains the following theorem [6,Thm. 1.5]: Theorem 1.1. (Rajala) Let Ω ⊂ R be a simply connected domain and ( X, d )a metric space of locally finite Hausdorff 2-measure. There exists a quasiconfor-mal homeomorphism f : X → Ω if and only if there exists a 2-quasiconformalhomeomorphism f : X → Ω.This result is proved using the measurable Riemann mapping theorem alongwith the classical John’s theorem on convex bodies. The latter theorem asserts,in part, that any convex body A in R n contains a unique ellipsoid E of maximalvolume satisfying E ⊂ A ⊂ √ nE , where the constant √ n is the best possible. Theconstant 2 in Theorem 1.1 is derived from the constant √ Theorem 1.2.
Let Ω ⊂ R be a simply connected domain and ( X, d ) a metricspace of locally finite Hausdorff 2-measure. There exists a quasiconformal homeo-morphism f : X → Ω if and only if there exists a quasiconformal homeomorphism f : X → Ω satisfying 2 π Mod Γ ≤ Mod f Γ ≤ π Mod Γ . (1)Rajala’s techniques, together with standard volume ratio estimates (see for in-stance [1, Thm. 6.2]), guarantee the existence of a quasiconformal map f O : X → Ωwith outer dilatation K O ≤ π/