Weil-Peterson curves β-numbers, and minimal surfaces

Abstract

This paper gives geometric characterizations of the Weil-Petersson class of rectifiable quasicircles, i.e., the closure of the smooth planar curves in the Weil-Petersson metric on universal Teichmüller space defined by Takhtajan and Teo. Although motivated by the planar case, many of our characterizations make sense for curves in Rn and remain equivalent in all dimensions. We prove that Γ is Weil-Petersson if and only if it is well approximated by polygons in a precise sense, has finite Möbius energy or has arclength parameterization in H3/2(T). Other results say that a curve is Weil-Petersson if and only if local curvature is square integrable over all locations and scales, where local curvature is measured using various quantities such as Jones’s β-numbers, nonlinearity of conformal weldings, Menger curvature, the “thickness” of the hyperbolic convex hull of Γ, and the total curvature of minimal surfaces in hyperbolic space. Finally, we prove that planar Weil Petersson curves are exactly the asymptotic boundaries of minimal surfaces in Hs with finite renormalized area.

Authors

Christopher J. Bishop

Mathematics Department, Stony Brook University, Stony Brook, NY 11794-3651