2,460 research outputs found
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
Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials
We study the two-point correlation functions for the zeroes of systems of
-invariant Gaussian random polynomials on and systems
of -invariant Gaussian analytic functions. Our result
reflects the same "repelling," "neutral," and "attracting" short-distance
asymptotic behavior, depending on the dimension, as was discovered in the
complex case by Bleher, Shiffman, and Zelditch. For systems of the -invariant Gaussian analytic functions we also obtain a
fast decay of correlations at long distances.
We then prove that the correlation function for the -invariant Gaussian analytic functions is "universal,"
describing the scaling limit of the correlation function for the restriction of
systems of the -invariant Gaussian random polynomials to any
-dimensional submanifold . This provides a
real counterpart to the universality results that were proved in the complex
case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the
complex case, proving a special case of the universality results of Bleher,
Shiffman, and Zelditch.)Comment: 28 pages, 1 figure. To appear in International Mathematics Research
Notices (IMRN
Andreev's Theorem on hyperbolic polyhedra
In 1970, E. M. Andreev published a classification of all three-dimensional
compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a
combinatorial description of a polyhedron, , Andreev's Theorem provides five
classes of linear inequalities, depending on , for the dihedral angles,
which are necessary and sufficient conditions for the existence of a hyperbolic
polyhedron realizing with the assigned dihedral angles. Andreev's Theorem
also shows that the resulting polyhedron is unique, up to hyperbolic isometry.
Andreev's Theorem is both an interesting statement about the geometry of
hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof
for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is
also remarkable to what level the proof of Andreev's Theorem resembles (in a
simpler way) the proof of Thurston.
We correct a fundamental error in Andreev's proof of existence and also
provide a readable new proof of the other parts of the proof of Andreev's
Theorem, because Andreev's paper has the reputation of being ``unreadable''.Comment: To appear les Annales de l'Institut Fourier. 47 pages and many
figures. Revision includes significant modification to section 4, making it
shorter and more rigorous. Many new references include
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