3,330 research outputs found
Lengths of simple loops on surfaces with hyperbolic metrics
Given a compact orientable surface of negative Euler characteristic, there
exists a natural pairing between the Teichmueuller space of the surface and the
set of homotopy classes of simple loops and arcs. The length pairing sends a
hyperbolic metric and a homotopy class of a simple loop or arc to the length of
geodesic in its homotopy class. We study this pairing function using the
Fenchel-Nielsen coordinates on Teichmueller space and the Dehn-Thurston
coordinates on the space of homotopy classes of curve systems. Our main result
establishes Lipschitz type estimates for the length pairing expressed in terms
of these coordinates. As a consequence, we reestablish a result of
Thurston-Bonahon that the length pairing extends to a continuous map from the
product of the Teichmueller space and the space of measured laminations.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper17.abs.htm
Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point
We derive explicit bounds for the remainder term in the local Weyl law for
locally hyperbolic manifolds, we also give the estimates of the derivative of
this remainder. We use these to obtain explicit bounds for the C^k-norms of the
L^2-normalised eigenfunctions in the case spectrum of the Laplacian is
discrete, e.g. for closed Riemannian manifolds. We also derive bounds for the
local heat trace. Our estimates are purely local and therefore also hold for
any manifold at points near which the metric is locally hyperbolic.Comment: 23 pages, 2 figure
Noneuclidean Tessellations and their relation to Reggie Trajectories
The coefficients in the confluent hypergeometric equation specify the Regge
trajectories and the degeneracy of the angular momentum states. Bound states
are associated with real angular momenta while resonances are characterized by
complex angular momenta. With a centrifugal potential, the half-plane is
tessellated by crescents. The addition of an electrostatic potential converts
it into a hydrogen atom, and the crescents into triangles which may have
complex conjugate angles; the angle through which a rotation takes place is
accompanied by a stretching. Rather than studying the properties of the wave
functions themselves, we study their symmetry groups. A complex angle indicates
that the group contains loxodromic elements. Since the domain of such groups is
not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of
the isometric circle is adapted since it treats all groups symmetrically. The
pairing of circles and their inverses is likened to pairing particles with
their antiparticles which then go one to produce nested circles, or a
proliferation of particles. A corollary to Laguerre's theorem, which states
that the euclidean angle is represented by a pure imaginary projective
invariant, represents the imaginary angle in the form of a real projective
invariant.Comment: 27 pages, 4 figure
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