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

• 成果类型: Article
• 署名作者: Bishop, Christopher J.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2025.202.1.2
• 发表日期: 2025
• 页码: 111-188
• 关键词: universal teichmuller space; traveling salesman problem; renormalized volume; univalent-functions; rectifiable curves; sharp inequality; string theory; convex hulls; dirichlet; energy

摘要

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 Teichmuller 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 Gamma is Weil-Petersson if and only if it is well approximated by polygons in a precise sense, has finite Mobius energy or has arclength parametrization 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 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 H3 with finite renormalized area.