The p-Weil–Petersson Teichmüller Space and the Quasiconformal Extension of Curves

Abstract

We consider the correspondence between the space of p-Weil–Petersson curves $$\gamma $$ γ on the plane and the p-Besov space of $$u=\log \gamma '$$ u = log γ ′ on the real line for $$p > 1$$ p > 1 . We prove that the variant of the Beurling–Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p-Besov space to the space of p-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of p-integrable Beltrami coefficients to the p-Weil–Petersson Teichmüller space.

Parametrization of the p-Weil–Petersson Curves: Holomorphic Dependence

“…Then, the element f in W p for p > 1 can be characterized by the property that f is locally absolutely continuous and log f belongs to the p-Besov space B R p (R) of real-valued functions, which coincides with H 1/2 R (R) for p = 2. In fact, in our paper [47] generalizing [36], it was proved that if a quasisymmetric homeomorphism f of R is locally absolutely continuous and log f is in B R p (R), then the variant of the Beurling-Ahlfors extension by the heat kernel introduced by Fefferman, Kenig and Pipher [10] yields a quasiconformal homeomorphism of U whose complex dilatation is in M p (U). We will elaborate these concepts in Sect.…”

“…This induces the inverse of the biholomorphic map L on the neighborhood U (B R p (R)) as shown in [47,Theorem 4.5].…”

“…We know that the p-Weil-Petersson Teichmüller space T p is contractible by [8,Theorem 6] for p = 2, by [52, Proposition 3.5] for p ≥ 2, and by [47,Corollary 1.4] and [50, Proposition 5.6] for general p > 1. (In fact, the above argument giving the identification T p ∼ = W p ∼ = B R p (R) for p > 1 produces Corollary 5.2 (1) below, which implies that T p is contractible.)…”

Chordal Loewner chains and Teichmüller spaces on the half-plane

“…The universal Teichmüller space T can be regarded as the set of all conformal mappings h (up to the composition of a Möbius transformation) on H which can be extended to a quasiconformal homeomorphism of C. Subspaces of T are obtained by imposing some conditions on h in general. If this condition is conformally invariant, then there is no difference between the cases where we consider it on D and on H. The Weil-Petersson Teichüller space and the BMO-Teichmüller space are such examples, which have received much attention over the years (see [3,13,35,36,39,40] and references therein).…”

The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by heat kernel

“…Corollary IV.5.15]), and moreover, the A ∞ -constant C ∞ (e ru ) for e ru depends only on u 0 (see also the proof of[31, Lemma 2.6]). By the property of A ∞ -weights in(9) and the Cauchy-Schwarz inequality, we have…”