Geometriae Dedicata最新文献

We consider Meissner polyhedra in (mathbb {R}^3). These are constant width bodies whose boundaries consist of pieces of spheres and spindle tori. We define these shapes by taking appropriate intersections of congruent balls and show that they are dense within the space of constant width bodies in the Hausdorff topology. This density assertion was essentially established by Sallee. However, we offer a modern viewpoint taking into consideration the recent progress in understanding ball polyhedra and in constructing constant width bodies based on these shapes.

In this paper we are interested in algebraic intersection of closed curves of a given length on translation surfaces. We study the quantity KVol, defined in Cheboui et al. (Bull Soc Math France 149(4):613–640, 2021) and studied in Cheboui et al. (2021), Cheboui et al. (C R Math Acad Sci Paris 359:65–70, 2021), Boulanger et al. (Ann Henri Lebesgue, 2024), and Boulanger (Algebraic intersection, lengths and Veech surfaces, 2023. arXiv:2309.17165), and we construct families of translation surfaces in each connected component of the minimal stratum (mathcal {H}(2g-2)) of the moduli space of translation surfaces of genus (g ge 2) such that KVol is arbitrarily close to the genus of the surface, which is conjectured to be the infimum of KVol on (mathcal {H}(2g-2)).

Let X be a compact connected Riemann surface and ((V,, phi )) a holomorphic Lie algebroid on X such that the holomorphic vector bundle V is stable. We give a necessary and sufficient condition on holomorphic vector bundles E on X to admit a Lie algebroid connection.

We describe an algorithm that takes as input an open book decomposition of a closed oriented 4-manifold and outputs an explicit trisection diagram of that 4-manifold. Moreover, a slight variation of this algorithm also works for open books on manifolds with non-empty boundary and for 3-manifold bundles over (S^1). We apply this algorithm to several simple open books, demonstrate that it is compatible with various topological constructions, and argue that it generalizes and unifies several previously known constructions.

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to (mathcal D_a)—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

Let (G_{6,3}) be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation (rho ) of (G_{6,3}) into (textbf{PU}(2,1)). We show the 3-orbifold at infinity of (rho (G_{6,3})) is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the ({mathbb {Z}}_3)-coned chain-link (C(6,-2)). This answers the second part of Kapovich’s Conjecture 10.6 in Kapovich (in: In the tradition of thurston II. Geometry and groups, Springer, Cham, 2022), and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz’s first example in Schwartz (Invent Math 151(2):221–295, 2003).

Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic varies through a confocal pencil and discover cases when each conic from the confocal family is inscribed in an n-polygon, which is inscribed in the circle, with the same n. Complete geometric characterization of such cases for (nin {4,6}) is given and proved that this cannot happen for other values of n. We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then ({text {diam}}(X) / {text {diam}}(X / G ) le 4 sqrt{ vert G vert }). For a group G and a finite symmetric generating set S, (P_k(varGamma (G, S))) denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph (varGamma ) of G with respect to S and whose 2-cells are m-gons for (0 le m le k), defined by the simple graph loops of length m in (varGamma ), up to cyclic permutations. Let G be a finite abelian group with (vert G vert ge 3) and S a symmetric set of generators for which (P_k(varGamma (G,S))) has trivial first Betti number. We show that the first nontrivial eigenvalue (-lambda _1) of the Laplacian on the Cayley graph satisfies (lambda _1 ge 2 - 2 cos ( 2 pi / k ) ). We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form (O (k^2 vert S vert log vert G vert )). Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained.

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

We study the geometry of the space of projectivized filling geodesic currents (mathbb {P}mathcal {C}_{fill}(S)). Bonahon showed that Teichmüller space, (mathcal {T}(S)) embeds into (mathbb {P}mathcal {C}_{fill}(S)). We extend the symmetrized Thurston metric from (mathcal {T}(S)) to the entire (projectivized) space of filling currents, and we show that (mathcal {T}(S)) is isometrically embedded into the bigger space. Moreover, we show that there is no quasi-isometric projection back down to (mathcal {T}(S)). Lastly, we study the geometry of a length-minimizing projection from (mathbb {P}mathcal {C}_{fill}(S)) to (mathcal {T}(S)) defined previously by Hensel and the author.