928 research outputs found
Minimal kernels of Dirac operators along maps
Let be a closed spin manifold and let be a closed manifold. For maps
and Riemannian metrics on and on , we consider
the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index
in . If is -dimensional, one gets a lower bound for
the dimension of the kernel of out of this index. We investigate
the question whether this lower bound is obtained for generic tupels
Manifolds with small Dirac eigenvalues are nilmanifolds
Consider the class of n-dimensional Riemannian spin manifolds with bounded
sectional curvatures and diameter, and almost non-negative scalar curvature.
Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of
the Dirac operator on such a manifold has small eigenvalues, then the
manifold is diffeomorphic to a nilmanifold and has trivial spin structure.
Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a
non-trivial spin structure, then there exists a uniform lower bound on the r-th
eigenvalue of the square of the Dirac operator. If a manifold with almost
nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume
is not too small, then we show that the metric is close to a Ricci-flat metric
on M with a parallel spinor. In dimension 4 this implies that M is either a
torus or a K3-surface
Fluid dynamics of bacterial turbulence
Self-sustained turbulent structures have been observed in a wide range of
living fluids, yet no quantitative theory exists to explain their properties.
We report experiments on active turbulence in highly concentrated 3D
suspensions of Bacillus subtilis and compare them with a minimal fourth-order
vector-field theory for incompressible bacterial dynamics. Velocimetry of
bacteria and surrounding fluid, determined by imaging cells and tracking
colloidal tracers, yields consistent results for velocity statistics and
correlations over two orders of magnitude in kinetic energy, revealing a
decrease of fluid memory with increasing swimming activity and linear scaling
between energy and enstrophy. The best-fit model parameters allow for
quantitative agreement with experimental data.Comment: 5 pages, 4 figure
Nonexistence of Generalized Apparent Horizons in Minkowski Space
We establish a Positive Mass Theorem for initial data sets of the Einstein
equations having generalized trapped surface boundary. In particular we answer
a question posed by R. Wald concerning the existence of generalized apparent
horizons in Minkowski space
Families index theory for Overlap lattice Dirac operator. I
The index bundle of the Overlap lattice Dirac operator over the orbit space
of lattice gauge fields is introduced and studied. Obstructions to the
vanishing of gauge anomalies in the Overlap formulation of lattice chiral gauge
theory have a natural description in this context. Our main result is a formula
for the topological charge (integrated Chern character) of the index bundle
over even-dimensional spheres in the orbit space. It reduces under suitable
conditions to the topological charge of the usual (continuum) index bundle in
the classical continuum limit (this is announced and sketched here; the details
will be given in a forthcoming paper). Thus we see that topology of the index
bundle of the Dirac operator over the gauge field orbit space can be captured
in a finite-dimensional lattice setting.Comment: Latex, 23 pages. v4: minor improvements, results unchanged; to appear
in Nucl.Phys.
Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms
We give a spinorial characterization of isometrically immersed surfaces of
arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For
Lorentzian surfaces, this generalizes a recent work of the first author in
to other Lorentzian space forms. We also characterize
immersions of Riemannian surfaces in these spaces. From this we can deduce
analogous results for timelike immersions of Lorentzian surfaces in space forms
of corresponding signature, as well as for spacelike and timelike immersions of
surfaces of signature (0,2), hence achieving a complete spinorial description
for this class of pseudo-Riemannian immersions.Comment: 9 page
Spectral Bounds for Dirac Operators on Open Manifolds
We extend several classical eigenvalue estimates for Dirac operators on
compact manifolds to noncompact, even incomplete manifolds. This includes
Friedrich's estimate for manifolds with positive scalar curvature as well as
the author's estimate on surfaces.Comment: pdflatex, 14 pages, 3 figure
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
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