Weil-Petersson curves, β-numbers, and minimal surfaces

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 parametrization in H³/²((equation presented)). 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 ℍ³ with finite renormalized area.