A Thurston boundary and visual sphere of the universal Teichmüller space (Q2046148)

The universal Teichmüller space \(\mathcal{T}(\mathbb{D})\) is the space of all quasisymmetric homeomorphisms of \(\partial \mathbb{D}=\mathbb{S}^1\) subject to a normalization condition (i.e., fixing 1, \(i\) and \(-1\) in \(\mathbb{S}^1\)). The universal Teichmüller space homeomorphically embeds in bounded geodesic current space as Liouville geodesic currents [\textit{F. Bonahon} and the second author, Math. Ann. 380, No. 3--4, 1119--1167 (2021; Zbl 1472.57016)], generalizing Bonahon's geodesic current theory for closed surfaces [\textit{F. Bonahon}, Invent. Math. 92, No. 1, 139--162 (1988; Zbl 0653.32022)]. The second author [Topology 44, No. 1, 99--130 (2005; Zbl 1082.30036)], and \textit{F. Bonahon} and the second author [Math. Ann. 380, No. 3--4, 1119--1167 (2021; Zbl 1472.57016)] used this setup to define a natural generalization of the Thurston bordification of Teichmüller space in terms of adding bounded projective measured laminations \(\mathcal{PML}_{bdd}(\mathbb{D})\) -- a space consisting of \(\mathbb{R}_+\)-rays of \textit{bounded geodesic currents} asymptotic to the embedding of \(\mathcal{T}(\mathbb{D})\). The primary aim of this paper is to establish the convergence, to \(\mathcal{PML}_{bdd}(\mathbb{D})\), of certain families of of Teichmüller metric geodesics in \(\mathcal{T}(\mathbb{D})\).\medskip \textit{H. Masur} [Duke Math. J. 49, 183--190 (1982; Zbl 0508.30039)] showed that for Teichmüller spaces \(\mathcal{T}(S)\) of closed surfaces \(S\), Teichmüller metric geodesic rays, obtained from shrinking the vertical foliations for integrable holomorphic quadratic differentials whose vertical foliations are either \begin{itemize} \item[1.] uniquely ergodic, or \item[2.] consist of finitely many cylinders, \end{itemize} necessarily converge to a single point on the Thurston boundary (i.e., the projective measured lamination space \(\mathcal{PML}(S)\) of \(\mathcal{T}(S)\)). Case~1 converges to the unique projective measured lamination corresponding to the vertical foliation, and case~2 converges to the projectively uniformly weighted multicurve corresponding to the vertical cylinders. Note, however, Teichmüller metric geodesics on \(\mathcal{T}(S)\) don't always asymptotically converge to a single point on \(\mathcal{PML}(S)\), see, e.g., [\textit{A. Lenzhen}, Geom. Topol. 12, No. 1, 177--197 (2008; Zbl 1189.30086); \textit{A. Lenzhen} et al., J. Mod. Dyn. 12, 261--283 (2018; Zbl 1464.32016)]. In ostensible contradiction to the closed surface theory, the authors showed in [Proc. Lond. Math. Soc. (3) 116, No. 6, 1599--1628 (2018; Zbl 1394.30029)] that Teichmüller metric geodesics in \(\mathcal{T}(\mathbb{D})\) given by shrinking the vertical foliations for any integrable holomorphic quadratic differential (such paths are called \textit{Teichmüller rays}) \textit{always} converge with respect to the weak* topology. This seeming paradox that the ``universal'' generalization of a theory should fail to exhibit behavior present in a ``specific'' version might be due to the following factors: \begin{itemize} \item[1.] the \textit{uniform weak* topology}, rather than the weak* topology, is the ``correct'' topology to consider on bounded geodesic current space, and convergence is not guaranteed for the uniform weak* topology; \item[2.] lifts, to \(\mathbb{D}\), of non-zero holomorphic quadratic differentials on closed surfaces fail to be integrable. \end{itemize} This second point suggests that Teichmüller metric geodesics obtained from shrinking the vertical foliations of potentially non-integrable holomorphic differentials (i.e., \textit{generalized Teichmüller rays}) warrant deeper investigation, and this is the first paper to systematically do so. Specifically, the authors show that when the Euclidean structure induced by a holomorphic differential \(\varphi\) on \(\mathbb{D}\) produces \begin{itemize} \item[1.] the \textit{infinite strip}: \([0,1]\times\mathbb{R}\subset\mathbb{R}^2\cong\mathbb{C}\); \item[2.] \textit{Strebel's chimney}: \(\{z\in\mathbb{C}\mid \Im(z)<0\text{ or }\Re(z)<1\}\); or \item[3.] \textit{general domains with chimneys}: \(\mathbb{R}^2\cong\mathbb{C}\) with countably vertical slits \(\{a\}\times\mathbb{R}_{\pm}\) or half-strips \((b,c)\times\mathbb{R}_{\pm}\) removed (in such a way that the singletons \(\{a\}\) and intervals \((b,c)\) in the first coordinates of these removed sets are discretely distributed over \(\mathbb{R}\)), \end{itemize} then the generalized Teichmüller rays for such an \(\varphi\) converge to (projective classes of) linear combinations of Dirac measures supported on geodesics in \(\mathbb{D}\) with endpoints placed at the points in \(\partial\mathbb{D}\) corresponding to either the various ``infinite ends'' or the boundary corners of the \(\varphi\)-induced Euclidean structures on \(\mathbb{D}\) (see the paper for explicit descriptions). Note that the authors establish weak* convergence in this paper, and it is not yet apparent to the reviewer whether uniform weak* convergence holds (although the results obtained are so reasonable that one might reasonable question the usefulness of uniform weak* convergence should such convergence fail, at least for the case when there are only finitely many removed slits and half-strips). In order to prove these convergence results, the authors convert the requisite Liouville current estimates needed to verify weak* convergence into modulus estimates (Lemma~3.7 and Corollary~3.10). Then, to obtain the behaviour of certain moduli with respect to deformation along a generalized Teichmüller ray, they establish upper and lower bounds for the aforementioned moduli in terms of classical moduli computations for curve families on rectangles and annuli sectors. The ``first-principles'' nature and the roughness of these estimates hints at room for generalization. It would be interesting to see further explorations along these themes (and perhaps via other types of geodesics for Teichmüller space, which illustrate similar convergence properties in the closed surface context, see [\textit{G. Théret}, Ann. Acad. Sci. Fenn., Math. 32, No. 2, 381--408 (2007; Zbl 1125.30040); Geom. Dedicata 136, 79--93 (2008; Zbl 1167.30027); \textit{R. Díaz} and \textit{C. Series}, Algebr. Geom. Topol. 3, 207--234 (2003; Zbl 1066.32020)], for example), and we hope that these investigations might in time suggest a less restrictive replacement for the integrability condition currently employed in universal Teichmüller theory.