9,047 research outputs found
The second Yamabe invariant
Let be a compact Riemannian manifold of dimension . We
define the second Yamabe invariant as the infimum of the second eigenvalue of
the Yamabe operator over the metrics conformal to and of volume 1. We study
when it is attained. As an application, we find nodal solutions of the Yamabe
equation
The supremum of conformally covariant eigenvalues in a conformal class
Let (M,g) be a compact Riemannian manifold of dimension >2. We show that
there is a metric h conformal to g and of volume 1 such that the first positive
eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A
similar statement is proven for the first positive eigenvalue of the Dirac
operator on a spin manifold of dimension >1
Relations between threshold constants for Yamabe type bordism invariants
In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe
invariant on closed manifolds is a bordism invariant below a certain threshold
constant. A similar result holds for a spinorial analogon. These threshold
constants are characterized through Yamabe-type equations on products of
spheres with rescaled hyperbolic spaces. We give variational characterizations
of these threshold constants, and our investigations lead to an explicit
positive lower bound for the spinorial threshold constants
The first conformal Dirac eigenvalue on 2-dimensional tori
Let M be a compact manifold with a spin structure \chi and a Riemannian
metric g. Let \lambda_g^2 be the smallest eigenvalue of the square of the Dirac
operator with respect to g and \chi. The \tau-invariant is defined as
\tau(M,\chi):= sup inf \sqrt{\lambda_g^2} Vol(M,g)^{1/n} where the supremum
runs over the set of all conformal classes on M, and where the infimum runs
over all metrics in the given class. We show that \tau(T^2,\chi)=2\sqrt{\pi} if
\chi is ``the'' non-trivial spin structure on T^2. In order to calculate this
invariant, we study the infimum as a function on the spin-conformal moduli
space and we show that the infimum converges to 2\sqrt{\pi} at one end of the
spin-conformal moduli space.Comment: published version (typos removed, bibliography updated
Function Spaces on Singular Manifolds
It is shown that most of the well-known basic results for Sobolev-Slobodeckii
and Bessel potential spaces, known to hold on bounded smooth domains in
, continue to be valid on a wide class of Riemannian manifolds
with singularities and boundary, provided suitable weights, which reflect the
nature of the singularities, are introduced. These results are of importance
for the study of partial differential equations on piece-wise smooth domains.Comment: 37 pages, 1 figure, final version, augmented by additional
references; to appear in Math. Nachrichte
Synthesis of electro-optic modulators for amplitude modulation of light
Electro-optical modulator realizes voltage transfer function in synthesizing birefringent networks. Choice of the voltage transfer function is important, the most satisfactory optimizes the modulator property
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