Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

3D B-Rep meshing for real-time data-based geometric parametric analysis

This paper presents an effective framework to automatically construct 3D quadrilateral meshes of complicated geometry and arbitrary topology adapted for parametric studies. The input is a triangulation of the solid 3D model’s boundary provided from B-Rep CAD models or scanned geometry. The triangulated mesh is decomposed into a set of cuboids in two steps: pants decomposition and cuboid decomposition. This workflow includes an integration of a geometry-feature-aware pants-to-cuboids decomposition algorithm. This set of cuboids perfectly replicates the input surface topology. Using aligned global parameterization, patches are re-positioned on the surface in a way to achieve low overall distortion, and alignment to principal curvature directions and sharp features. Based on the cuboid decomposition and global parameterization, a 3D quadrilateral mesh is extracted. For different parametric instances with the same topology but different geometries, the MEG-IsoQuad method allows to have the same representation: isotopological meshes holding the same connectivity where each point on a mesh has an analogous one into all other meshes. Faithful 3D numerical charts of parametric geometries are then built using standard data-based techniques. Geometries are then evaluated in real-time. The efficiency and the robustness of the proposed approach are illustrated through a few parametric examples.

Teichmuller Space

This chapter deals with Teichmüller space Teich(S) of a surface S. It first defines Teichmüller space and a topology on Teich(S) before giving two heuristic counts of its dimension. It then describes explicit coordinates on Teich(Sɡ) coming from certain length and twist parameters for curves in a pair of pants decomposition of Sɡ; these are the Fenchel–Nielsen coordinates on Teich(Sɡ). The chapter also considers the Teichmüller space of the torus and concludes by proving the 9g – 9 theorem, which states that a hyperbolic structure on Sɡ is completely determined by the lengths assigned to 9g – 9 isotopy classes of simple closed curves in Sɡ.

Fenchel–Nielsen coordinates on upper bounded pants decompositions

Let X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

A Short Note on Short Pants

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied, and the best upper bounds to date are linear in genus, due to a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound that slightly improves the best known bound.

BERS' CONSTANTS FOR PUNCTURED SPHERES AND HYPERELLIPTIC SURFACES

The main goal of this paper is to present a proof of Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main result states that any hyperbolic sphere with n cusps has a pants decomposition with all of its geodesics of length bounded by [Formula: see text]. Other results include lower and upper bounds for Bers' constants for hyperelliptic surfaces and spheres with boundary geodesics.