Dirac eigenspinors for generic metrics

Abstract

We consider a Riemannian spin manifold (M,g) with a fixed spin structure. The zero sets of solutions of generalized Dirac equations on M play an important role in some questions arising in conformal spin geometry and in mathematical physics. In this setting the mass endomorphism has been defined as the constant term in an expansion of Green's function for the Dirac operator. One is interested in obtaining metrics, for which it is not zero. In this thesis we study the dependence of the zero sets of eigenspinors of the Dirac operator on the Riemannian metric. We prove that on closed spin manifolds of dimension 2 or 3 for a generic Riemannian metric the non-harmonic eigenspinors have no zeros. Furthermore we prove that on closed spin manifolds of dimension 3 the mass endomorphism is not zero for a generic Riemannian metric.

PhD thesis, 96 pages

References in corpus (2)

• Mass endomorphism, surgery and perturbations
• Witten spinors on maximal, conformally flat hypersurfaces

Cited by in corpus (8)

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• New spinorial approach to mass inequalities for black holes in general relativity
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• An estimate on the nodal set of eigenspinors on closed surfaces
• Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds