448 research outputs found

    Conical Existence of Closed Curves on Convex Polyhedra

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    Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of "source unfolding" of the entire surface of P to one non-overlapping piece, as reported in a companion paper.Comment: 24 pages, 15 figures, 6 references. Version 2 includes a solution to one of the open problems posed in Version 1, concerning quasigeodesic loop

    Collisions of particles in locally AdS spacetimes I. Local description and global examples

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    We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph Γ\Gamma. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than 2π2\pi on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of Γ\Gamma). We then adapt to this setting some notions like global hyperbolicity which are natural for Lorentz manifolds, and construct some examples of globally hyperbolic AdS manifolds with interacting particles.Comment: This is a rewritten version of the first part of arxiv:0905.1823. That preprint was too long and contained two types of results, so we sliced it in two. This is the first part. Some sections have been completely rewritten so as to be more readable, at the cost of slightly less general statements. Others parts have been notably improved to increase readabilit

    Ramification conjecture and Hirzebruch's property of line arrangements

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    The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a CAT[0] ramification and prove this in several cases. In the latter case we prove that the ramification is CAT[0] if the metric is non-negatively curved. We deduce that complex line arrangements in the complex projective plane studied by Hirzebruch have aspherical complement.Comment: 19 pages 1 figur

    Polyhedral hyperbolic metrics on surfaces

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    Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space H3{\mathbb{H}^{3}} and a group G of isometries of H3{\mathbb{H}^{3}} such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedr
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