Function theoretic characterizations of Weil–Petersson curves

Abstract

The Weil-Petersson class is the closure of the smooth closed curves in the Weil-Petersson metric on universal Teichmüller space defined by Takhtajan and Teo. We give some new characterizations of this class of curves and some new proofs of previously known characterizations. In particular, we give a new, more geometric characterization of the conformal weldings of such curves and characterize the curves themselves in terms of Peter Jones's ˇ-numbers."Busca una situación en la que tu trabajo te dé tanta felicidad como tu tiempo libre." -Picasso. For Antonio Córdoba and José Luis Fernández. I hope they enjoyed discovering their results as much as the rest of us enjoyed reading about them.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

“…Since Möbius transformations of $$\hat{\mathbb {C}}$$ extend to isometries of the hyperbolic 3‐space $$\mathbb {H}^3$$ (whose boundary at $$\infty$$ is identified with the Riemann sphere $$\hat{\mathbb {C}}$$) and being a Weil–Petersson quasicircle is a Möbius invariant property, it is natural to try to relate our foliations by Weil–Petersson quasicircles to objects in $$\mathbb {H}^3$$. In [2], it is shown that every Jordan curve bounds at least one minimal disk in $$\mathbb {H}^3$$, and [8] shows that a Jordan curve is a Weil–Petersson quasicircle if and only if any such minimal disk in $$\mathbb {H}^3$$ has finite total curvature. For example, when $$\gamma$$ is a circle, the unique minimal surface is the totally geodesic surface, namely, the hemisphere bounded by $$\gamma$$.…”

“…A Jordan curve $$\gamma$$ has finite Loewner energy if and only if it belongs to the set of Weil–Petersson quasicircles [61], a class of nonsmooth chord–arc Jordan curves that has a number of equivalent characterizations from various different perspectives, see, for example, [8, 12, 28, 56, 57, 61, 65]. One way to characterize Weil–Petersson quasicircles is to say that their welding homeomorphisms $$h^{-1} \circ f|_{S^1}$$ (more precisely, the equivalence class modulo left‐action by the group of Möbius transformations preserving $$S^1$$) belong to the Weil–Petersson Teichmüller space $$T_0(1)$$, defined as the completion of $${\rm M\ddot{o}b}(S^1) \backslash \operatorname{Diff}^\infty (S^1)$$ using its unique homogeneous Kähler metric.…”

The Loewner Energy via the Renormalised Energy of Moving Frames

“…In particular, the class of finite energy curves corresponds exactly to the Weil-Petersson class of quasicircles which has already been extensively studied by both physicists and mathematicians since the eighties, see, e.g., [6,10,18,22,24,30,39,40,45,48,49,53,55,56,61], and is still an active research area. See the introduction of [6] (see also the companion papers [7,8] for more on this topic) for a summary and a list of equivalent definitions of very different nature.…”

“…Theorem 1.1. (Cui,[18], Tahktajan-Teo, [53], Shen, [49], Bishop, [6]) Let γ ⊂ C be a Jordan curve, be the bounded connected component of C \ γ , and let f : D → and g : C \ D → C \ be biholomorphic maps such that g(∞) = ∞.…”

Drivers, hitting times, and weldings in Loewner's equation

“…The Loewner energy was introduced (without the name) in [10] and subsequently saw rapid development in [31,[43][44][45][46], to give an incomplete list. In short, it has fascinating connections to a diverse array of fields: probability theory, complex analysis, hyperbolic geometry, geometric measure theory, and Teichmüller theory (see, e.g., [5,6,40]). See [47] for a helpful overview.…”