418 research outputs found
Enlargeability, foliations, and positive scalar curvature
We extend the deep and important results of Lichnerowicz, Connes, and
Gromov-Lawson which relate geometry and characteristic numbers to the existence
and non-existence of metrics of positive scalar curvature (PSC). In particular,
we show: that a spin foliation with Hausdorff homotopy groupoid of an
enlargeable manifold admits no PSC metric; that any metric of PSC on such a
foliation is bounded by a multiple of the reciprocal of the foliation K-area of
the ambient manifold; and that Connes' vanishing theorem for characteristic
numbers of PSC foliations extends to a vanishing theorem for Haefliger
cohomology classes.Comment: To appear in Inventiones Mathematicae. We have made a minor editing
chang
An interesting example for spectral invariants
In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a
Bismut superconnection for a family of generalized Dirac operators is defined
along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin
invariants of the Dirac operators were assumed greater than three times the
codimension of the foliation. It was then showed that the associated heat
operator converges to the Chern character of the index bundle of the operator.
In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing
the requirement on the Novikov-Shubin invariants to one half of the
codimension. In this paper, we construct examples which show that this is the
best possible result.Comment: Third author added. Some typos corrected and some material added.
Appeared in Journal of K Theory, Volume 13, in 2014, pages 305 to 31
Gerbes, simplicial forms and invariants for families of foliated bundles
The notion of a gerbe with connection is conveniently reformulated in terms
of the simplicial deRham complex. In particular the usual Chern-Weil and
Chern-Simons theory is well adapted to this framework and rather easily gives
rise to `characteristic gerbes' associated to families of bundles and
connections. In turn this gives invariants for families of foliated bundles. A
special case is the Quillen line bundle associated to families of flat
SU(2)-bundlesComment: 28 page
Magnetized Non-linear Thin Shell Instability: Numerical Studies in 2D
We revisit the analysis of the Non-linear Thin Shell Instability (NTSI)
numerically, including magnetic fields. The magnetic tension force is expected
to work against the main driver of the NTSI -- namely transverse momentum
transport. However, depending on the field strength and orientation, the
instability may grow. For fields aligned with the inflow, we find that the NTSI
is suppressed only when the Alfv\'en speed surpasses the (supersonic)
velocities generated along the collision interface. Even for fields
perpendicular to the inflow, which are the most effective at preventing the
NTSI from developing, internal structures form within the expanding slab
interface, probably leading to fragmentation in the presence of self-gravity or
thermal instabilities. High Reynolds numbers result in local turbulence within
the perturbed slab, which in turn triggers reconnection and dissipation of the
excess magnetic flux. We find that when the magnetic field is initially aligned
with the flow, there exists a (weak) correlation between field strength and gas
density. However, for transverse fields, this correlation essentially vanishes.
In light of these results, our general conclusion is that instabilities are
unlikely to be erased unless the magnetic energy in clouds is much larger than
the turbulent energy. Finally, while our study is motivated by the scenario of
molecular cloud formation in colliding flows, our results span a larger range
of applicability, from supernovae shells to colliding stellar winds.Comment: 12 pages, 17 figures, some of them at low resolution. Submitted to
ApJ, comments welcom
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