136 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
Harmonic spinors and local deformations of the metric
Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index
theorem yields a lower bound for the dimension of the kernel of the Dirac
operator. We prove that this bound can be attained by changing the Riemannian
metric g on an arbitrarily small open set.Comment: minor changes, to appear in Mathematical Research Letter
The conformal Yamabe constant of product manifolds
Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3.
We derive a lower bound for the conformal Yamabe constant of the product
manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and
(W,h).Comment: 12 pages, to appear in Proc. AMS; v3: small changes, very last
preprint version, close to published versio
Low-dimensional surgery and the Yamabe invariant
Assume that M is a compact n-dimensional manifold and that N is obtained by
surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants
\sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for
\Lambda>0. We derive explicit lower bounds for \Lambda in dimensions where
previous methods failed, namely for (n,k)\in
{(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and
bordism theory several gap phenomena for smooth Yamabe invariants can be
deduced.Comment: Version 2 contains new results: the case (n,k)=(6,3) is now solved,
Version 3: typos corrected, final version to appear in J. Math. Soc. Japa
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