Recent zbMATH articles in MSC 30F40https://www.zbmath.org/atom/cc/30F402022-05-16T20:40:13.078697ZWerkzeugOn the discreteness of states accessible via right-angled paths in hyperbolic spacehttps://www.zbmath.org/1483.300832022-05-16T20:40:13.078697Z"Lessa, Pablo"https://www.zbmath.org/authors/?q=ai:lessa.pablo"Garcia, Ernesto"https://www.zbmath.org/authors/?q=ai:garcia.ernestoSummary: We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance \(r>0\) along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of \(r>0\) for which the set of orthonormal frames accessible using these transformations is discrete.
In the hyperbolic plane this is equivalent to solving the discreteness problem (see [\textit{J.Gilman}, Geom. Dedicata 201, 139--154 (2019; Zbl 1421.30056)] and the references therein) for a particular one parameter family of two-generator subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometryhttps://www.zbmath.org/1483.300842022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Danciger, Jeffrey"https://www.zbmath.org/authors/?q=ai:danciger.jeffrey"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Schlenker, Jean-Marc"https://www.zbmath.org/authors/?q=ai:schlenker.jean-marcA theorem by Alexandrov and Pogorelov says that any smooth Riemannian metric on the 2-sphere with curvature \(K>-1\) coincides with the induced metric on the boundary of some compact convex subset of hyperbolic 3-space with smooth boundary and, furthermore, that this compact convex subset is unique up to a global isometry of hyperbolic 3-space. In the paper under review, the authors study a generalization of this result to unbounded convex subsets of hyperbolic 3-space, more especially to convex subsets bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, they supplement the notion of induced metric on the boundary of the convex set so that it includes a gluing map at infinity which records how the asymptotic geometries of the two surfaces fit together near the limiting Jordan curve. They restrict their study to the case where the induced metrics on the two bounding surfaces have constant curvature \(K \in [-1, 0)\) and were the Jordan curve at infinity is a quasicircle. In this case the gluing map becomes a quasisymmetric homeomorphism of the circle and the authors prove that for \(K\) in the given interval, any quasisymmetric map can be obtained as the gluing map at infinity along some quasicircle. They also obtain Lorentzian analogous of these results, in which hyperbolic 3-space is replaced by the 3-dimensional anti-de Sitter space \(\mathbb{A}d\mathbb{S}^3\), whose natural boundary is the Einstein space \(\mathrm{Ein}^{1,1}\), a conformal Lorentzian analogue of the Riemannian sphere. The authors say that their results may be viewed as universal versions of a conjecture of Thurston about the realization of metrics on boundaries of convex cores of quasifuchsian hyperbolic manifolds and of an analogue of this conjecture, due to Mess, in the setting of globally hyperbolic anti-de Sitter spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)