448 research outputs found
Conical Existence of Closed Curves on Convex Polyhedra
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
We investigate 3-dimensional globally hyperbolic AdS manifolds containing
"particles", i.e., cone singularities along a graph . We impose
physically relevant conditions on the cone singularities, e.g. positivity of
mass (angle less than 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 ).
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
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
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 and a group G of isometries of 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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