Shapes of tetrahedra with prescribed cone angles

Abstract

Abstract. Given real numbers 4π > θ 0 ≥ θ 1 ≥ θ 2 ≥ θ 3 > 0 so that 3 j=0 θ j = 4π, we provide a detailed description of the space of flat metrics on the 2-sphere with 4 conical points of cone angles θ 0 , θ 1 , θ 2 , θ 3 , endowed with a geometric structure arising from the area function.

Signature Calculation of Area Hermitian Form on Some Spaces of Polygons

“…Note that the calculations we do here are compatible with the ones in [11]. Also, the calculation of the signature when n = 3 is given in [3].…”

Moduli Spaces of Flat Tori with Prescribed Holonomy

“…Among the papers deeply relying on it about the theory of conical flat structures on the Riemann sphere, one can mention [Web93], [Par06], [GLL11], [BP15] and [Pas16] where some particular cases are considered in detail. The recent paper [McM17] deserves to be mentioned as well: in it, the author gives a more detailed treatment of the notion of cone-manifold than in [Thu98] and obtains a nice version of the Gauß-Bonnet theorem for complex hyperbolic cone-manifolds that he eventually uses to compute the volumes of the Picard/Deligne-Mostow/Thurston's moduli spaces.…”

Quadrangulations of sphere and ball quotients

“…In the paper cited above, Thurston constructs a more populous list of discrete groups acting on complex hyperbolic spaces, then those that classify triangulations. Although this list contains the list obtained in the paper by Deligne and Mostow on hypergeometric functions in several variables, the two papers are technically disjoint and several papers have been published to reconcile them (see for example Troyanov's paper , or Fillastre's article, there is also a recent work of González and López‐López dedicated exclusively to tetrahedra). Our first task in the current work is to provide another link, by showing that Thurston's space of cocyles is isomorphic to the cohomology group of an associated locally constant sheaf on the sphere in Deligne‐Mostow's paper (Theorem ).…”