The Dirac operator on generalized Taub-NUT spaces

TL;DR

This paper establishes conditions under which harmonic L^2 spinors are absent on certain spin manifolds, including generalized Taub-NUT spaces, confirming a previous conjecture and broadening understanding of geometric analysis on these manifolds.

Contribution

It provides sufficient conditions for the absence of harmonic L^2 spinors on manifolds with cone bundle structures, including generalized Taub-NUT metrics, extending prior conjectures.

Findings

Conditions for absence of harmonic L^2 spinors on cone bundle manifolds.

Verification of these conditions for generalized Taub-NUT metrics.

Proof of a conjecture by Vișinescu and the second author.

Abstract

We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author.