Weil-Petersson Curves, Conformal Energies, Β-Numbers, and Minimal Surfaces
WEIL-PETERSSON CURVES, CONFORMAL ENERGIES, β-NUMBERS, AND MINIMAL SURFACES CHRISTOPHER J. BISHOP 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 Teichm¨ullerspace defined by Takhtajan and Teo [125]. 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 M¨obiusenergy or has arclength parameterization in H3=2(T). Other re- sults say that a curve is Weil-Petersson if and only if local curvature is square inte- grable over all locations and scales, where local curvature is measured using various quantities such as Peter Jones's β-numbers, conformal welding, Menger curvature, the \thickness" of the hyperbolic convex hull of Γ, and the total curvature of min- imal 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. Date: June 27, 2019; revised May 22, 2020. 1991 Mathematics Subject Classification. Primary: 30C62, 30F60 Secondary: 53A10, 46E35, 60J67 . Key words and phrases. universal Teichm¨ullerspace, Weil-Petersson class, fractional Sobolev spaces, chord-arc curves, beta numbers, Schwarzian derivatives, Dirichlet class, minimal surfaces, renormalized area, finite total curvature, M¨obiusenergy, Loewner energy. The author is partially supported by NSF Grant DMS 1906259. 1 WEIL-PETERSSON, CONFORMAL ENERGY, β-NUMBERS, & MINIMAL SURFACES 1 1.
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