20 research outputs found
Non-rigidity of spherical inversive distance circle packings
We give a counterexample of Bowers-Stephenson's conjecture in the spherical
case: spherical inversive distance circle packings are not determined by their
inversive distances.Comment: 6 pages, one pictur
Compact hyperbolic tetrahedra with non-obtuse dihedral angles
Given a combinatorial description of a polyhedron having edges, the
space of dihedral angles of all compact hyperbolic polyhedra that realize
is generally not a convex subset of \cite{DIAZ}. If has five
or more faces, Andreev's Theorem states that the corresponding space of
dihedral angles obtained by restricting to {\em non-obtuse} angles is a
convex polytope. In this paper we explain why Andreev did not consider
tetrahedra, the only polyhedra having fewer than five faces, by demonstrating
that the space of dihedral angles of compact hyperbolic tetrahedra, after
restricting to non-obtuse angles, is non-convex. Our proof provides a simple
example of the ``method of continuity'', the technique used in classification
theorems on polyhedra by Alexandrow \cite{ALEX}, Andreev \cite{AND}, and
Rivin-Hodgson \cite{RH}.Comment: 19 page
Thurston's sphere packings on 3-dimensional manifolds, I
Thurston's sphere packing on a 3-dimensional manifold is a generalization of
Thusrton's circle packing on a surface, the rigidity of which has been open for
many years. In this paper, we prove that Thurston's Euclidean sphere packing is
locally determined by combinatorial scalar curvature up to scaling, which
generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere
packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity
that Thurston's Euclidean sphere packing can not be deformed (except by
scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife
Some open questions on anti-de Sitter geometry
We present a list of open questions on various aspects of AdS geometry, that
is, the geometry of Lorentz spaces of constant curvature -1. When possible we
point out relations with homogeneous spaces and discrete subgroups of Lie
groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open
questions. 19 page
EXAMPLES OF RIGID AND FLEXIBLE SEIFERT FIBRED CONE-MANIFOLDS
The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one, which are both Seifert fibre