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 $\mathbb{R}^n$ and remain equivalent in all dimensions. We prove that $\Gamma$ 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 $H^{3/2}(\mathbb{T})$. 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 $\beta$-numbers, nonlinearity of conformal weldings, Menger curvature, the “thickness” of the hyperbolic convex hull of $\Gamma$, 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 $\mathbb{H}^s$ with finite renormalized area.
