14,404 research outputs found
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant
under the action of a Fuchsian group of isometries (i.e. a group of isometries
leaving globally invariant a totally geodesic surface, on which it acts
cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric
to a hyperbolic metric with conical singularities of positive singular
curvature on a compact surface of genus greater than one. We prove that these
metrics are actually realised by exactly one convex Fuchsian polyhedron (up to
global isometries). This extends a famous theorem of A.D. Alexandrov.Comment: Some little corrections from the preceding version. To appear in Les
Annales de l'Institut Fourie
Fuchsian polyhedra in Lorentzian space-forms
Let S be a compact surface of genus >1, and g be a metric on S of constant
curvature K\in\{-1,0,1\} with conical singularities of negative singular
curvature. When K=1 we add the condition that the lengths of the contractible
geodesics are >2\pi. We prove that there exists a convex polyhedral surface P
in the Lorentzian space-form of curvature K and a group G of isometries of this
space such that the induced metric on the quotient P/G is isometric to (S,g).
Moreover, the pair (P,G) is unique (up to global isometries) among a particular
class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of
A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus
cases, and it is also the polyhedral version of a theorem of
Labourie--Schlenker
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
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