392 research outputs found
Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds
We show that the hyperbolic structure on a closed, orientable, hyperbolic
3-manifold can be constructed from a solution to the hyperbolic gluing
equations using any triangulation with essential edges. The key ingredients in
the proof are Thurston's spinning construction and a volume rigidity result
attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we
show that this gives a new algorithm to detect hyperbolic structures on closed
3-manifolds.Comment: 17 page
Curve diffusion and straightening flows on parallel lines
In this paper, we study families of immersed curves
with free boundary supported
on parallel lines
evolving by the curve diffusion flow and the curve straightening flow. The
evolving curves are orthogonal to the boundary and satisfy a no-flux condition.
We give estimates and monotonicity on the normalised oscillation of curvature,
yielding global results for the flows.Comment: 35 pages, 3 figure
Some remarks on the simplicial volume of nonpositively curved manifolds
We show that any closed manifold with a metric of nonpositive curvature that
admits either a single point rank condition or a single point curvature
condition has positive simplicial volume. We use this to provide a differential
geometric proof of a conjecture of Gromov in dimension three.Comment: 14 pages, 1 figure. Minor revision
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