Strong Szegő Theorem on a Jordan Curve

The Loewner Energy via the Renormalised Energy of Moving Frames

“…The second author proved in [59] that the Loewner energy is proportional to the universal Liouville action introduced by L. A. Takhtajan and L.-P. Teo [53]. 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.…”

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

“…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.…”