Geometric and analytic properties of families of hypersurfaces in Eguchi-Hanson space

Abstract

We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $\L^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infima of the essential spectra of the Laplace and of the square of the Dirac operator in the case of closed curves. For circles we also compute the $\L^2$-kernel of the Dirac operator in the sense of spectral theory and show that it is infinite dimensional. We consider further the Einstein Dirac system on these spaces and construct explicit examples of T-Killing spinors on them.