711 research outputs found
Long-time analysis of 3 dimensional Ricci flow III
In this paper we analyze the long-time behavior of 3 dimensional Ricci flows
with surgery. Our main result is that if the surgeries are performed correctly,
then only finitely many surgeries occur and after some time the curvature is
bounded by . This result confirms a conjecture of Perelman. In the
course of the proof, we also obtain a qualitative description of the geometry
as .
This paper is the third part of a series. Previously, we had to impose a
certain topological condition to establish the finiteness of
the surgeries and the curvature control. The objective of this paper is to
remove this condition and to generalize the result to arbitrary closed
3-manifolds. This goal is achieved by a new area evolution estimate for minimal
simplicial complexes, which is of independent interest.Comment: 86 page
Equilibration and Approximate Conservation Laws: Dipole Oscillations and Perfect Drag of Ultracold Atoms in a Harmonic Trap
The presence of (approximate) conservation laws can prohibit the fast
relaxation of interacting many-particle quantum systems. We investigate this
physics by studying the center-of-mass oscillations of two species of fermionic
ultracold atoms in a harmonic trap. If their trap frequencies are equal, a
dynamical symmetry (spectrum generating algebra), closely related to Kohn's
theorem, prohibits the relaxation of center-of-mass oscillations. A small
detuning of the trap frequencies for the two species breaks the
dynamical symmetry and ultimately leads to a damping of dipole oscillations
driven by inter-species interactions. Using memory-matrix methods, we calculate
the relaxation as a function of frequency difference, particle number,
temperature, and strength of inter-species interactions. When interactions
dominate, there is almost perfect drag between the two species and the
dynamical symmetry is approximately restored. The drag can either arise from
Hartree potentials or from friction. In the latter case (hydrodynamic limit),
the center-of-mass oscillations decay with a tiny rate, , where is a single particle scattering rate.Comment: 9 pages + 5 pages of appendix, 9 figures; changes in v2: updated
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