Small eigenvalues of the Conformal Laplacian

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the $\alpha$-genus.Comment: Remark 3.3 added. To appear in "Geometric And Functional Analysis

A density theorem for asymptotically hyperbolic initial data satisfying the dominant energy condition

When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, we show that an asymptotically hyperbolic initial data set with non-negative local energy density can be approximated by an initial data set with strictly positive local energy density and a simple structure at infinity, while changing the mass arbitrarily little. The argument follows an argument used by Eichmair, Huang, Lee, and Schoen in the asymptotically Euclidean case

Prescribing eigenvalues of the Dirac operator

In this note we show that every compact spin manifold of dimension $\geq 3$ can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.Comment: To appear in Manuscripta Mathematic

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

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