Chord-arc surfaces with small constant. II. Good parameterizations

AbstractIn a companion paper we introduced a notion for hypersurfaces in Rd + 1 that is analogous to the chord-arc condition with small constant for planar curves. In this paper we address the problem of finding good parameterizations for these surfaces. In particular, we show that these surfaces must be homeomorphic to Rd, and we build parameterizations with Lp bounds on the gradient, and the gradient of the inverse, with p → ∞ as the chord-arc constant goes to 0. When d = 2 we use the uniformization theorem to find a quasisymmetric parameterization. We do not know how to build bilipschitz parameterizations when d ≥ 2, or quasisymmetric parameterizations when d ≥ 3. The author conjectures that they do not exist in general. However, we can build bilipschitz reflections in all dimensions, and these reflections will satisfy certain extra estimates, in terms of Carleson measures