Shen, Yuliang; Tang, Shuan Weil-Petersson Teichmüller space. II: Smoothness of flow curves of \(H^{\frac{3}{2}} \)-vector fields. (English) Zbl 1436.30015 Adv. Math. 359, Article ID 106891, 25 p. (2020); corrigendum ibid. 399, Article ID 108015, 4 p. (2022). The universal Teichmüller space \(T = \text{QS}(S^1)/\text{Mob}(S^1)\) is the group of quasisymmetric homeomorphisms of the unit circle \(S^1\) up to a normalization by a Möbius map. The theory of Ahlfors-Bers shows that \(T\) has a structure of a complex Banach manifold, and any finite-dimensional Teichmüller space admits a holomorphic embedding in \(T\). The tangent space of \(T\) was shown to comprise continuous vector fields on \(S^1\) that are in a classical Zygmund space \(\Lambda_\ast\) with an additional normalization, by H. M. Reimann [Invent. Math. 33, 247–270 (1976; Zbl 0328.30019)]. However, the formal formula for the Weil-Petersson metric on \(T\), proposed by S. Nag and A. Verjovsky [Commun. Math. Phys. 130, No. 1, 123–138 (1990; Zbl 0705.32013)], converges only for those vector fields of Sobolev class \(H^{3/2}\). To overcome this difficulty, L. A. Takhtajan and L.-P. Teo [Weil-Petersson metric on the universal Teichmüller space. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1243.32010)] equipped \(T\) with a different complex Hilbert manifold structure where the formula for the Weil-Petersson metric converges. In the new topology, \(T\) has uncountably many connected components, and the connected component \(T_0\) containing the identity map is of special interest, since it is the Weil-Petersson completion of the (normalized) smooth orientation-preserving diffeomorphisms of \(S^1\). It is known that the tangent space of \(T_0\) at the identity consists of precisely the (normalized) \(H^{3/2}\) vector fields on \(S^1\). If we write \(T_0 = \text{WP}(S^1)/\text{Mob}(S^1)\), it is an interesting (and open) problem to give an intrinsic characterization of those quasisymmetric maps of \(S^1\) that lie in \(\text{WP}(S^1)\).Previous work of the first author [Am. J. Math. 140, No. 4, 1041–1074 (2018; Zbl 1421.30059)] shows that flows of \(H^{3/2}\)-vector fields on \(S^1\) always lie in \(\text{WP}(S^1)\). That is, if \(\lambda \in C^0([0,M], H^{3/2}(S^1))\) then the solution \(\eta(t, x)\) to the differential equation \[ \begin{cases} \frac{d\eta}{dt}=\lambda(t, \eta(t,x))\\ \eta(0,x)=x \end{cases} \] satisfies the property that \(\eta(t, \cdot) \in\text{WP}(S^1)\) for each \(t\in [0,M]\). In this paper, the authors prove that the mapping \(t\mapsto \eta(t, \cdot)\) is in fact continuously differentiable under the Hilbert manifold structure of Takhatajan-Teo. This proves a conjecture of F. Gay-Balmaz and T. S. Ratiu [Adv. Math. 279, 717–778 (2015; Zbl 1320.32018].Note that \(3/2\) is the critical exponent for the Sobolev embedding theorem in dimension \(1\); for any \(s>3/2\), the group \(\text{Diff}_+^s(S^1)\) of \(H^s\)-diffeomorphisms of \(S^1\) is a Hilbert manifold modelled on the vector fields of class \(H^s\) on \(S^1\), and contains exactly the flows of the time-dependent vector fields \(\lambda \in C^0([0,M], H^{s}(S^1))\). Reviewer: Subhojoy Gupta (Bangalore) Cited in 2 ReviewsCited in 12 Documents MSC: 30C62 Quasiconformal mappings in the complex plane 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30H30 Bloch spaces 30H35 BMO-spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:universal Teichmüller space; Weil-Petersson Teichmüller space; quasi-symmetric homeomorphism; quasiconformal mapping; Sobolev class; non-smooth flow Citations:Zbl 0328.30019; Zbl 0705.32013; Zbl 1243.32010; Zbl 1421.30059; Zbl 1320.32018 PDFBibTeX XMLCite \textit{Y. Shen} and \textit{S. Tang}, Adv. Math. 359, Article ID 106891, 25 p. (2020; Zbl 1436.30015) Full Text: DOI arXiv References: [1] Ahlfors, L. V., Lectures on Quasiconformal Mappings (1966), D. Van Nostrand: D. Van Nostrand Princeton, New York · Zbl 0138.06002 [2] Ahlfors, L. 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