[Paper Review] Dirac eigenspinors for generic metrics

Andreas Hermann|arXiv (Cornell University)|Jan 27, 2012

Spectral Theory in Mathematical Physics46 references9 citations

TL;DR

This paper establishes that on closed spin manifolds of dimension 2 or 3, non-harmonic eigenspinors of the Dirac operator have no zeros for a generic Riemannian metric. It further proves that the mass endomorphism is non-zero for a generic metric in dimension 3, resolving key questions in conformal spin geometry and mathematical physics related to zero sets and spectral invariants of the Dirac operator.

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.

Motivation & Objective

• To investigate the zero sets of eigenspinors of the Dirac operator on closed spin manifolds under variation of the Riemannian metric.
• To determine whether the mass endomorphism—the constant term in the Green's function expansion of the Dirac operator—can be non-zero for generic metrics.
• To establish genericity results for the absence of zeros in non-harmonic eigenspinors and non-vanishing mass endomorphism in low-dimensional spin geometry.
• To provide a foundation for spectral bounds and existence of nowhere-vanishing spinors relevant to the positive energy theorem and conformal invariants.

Proposed method

• Utilizes transversality theory and analytic perturbation techniques for families of Riemannian metrics.
• Applies holomorphic family methods to extend the Dirac operator along real-analytic one-parameter families of metrics.
• Employs the concept of a holomorphic family of type (A) to ensure spectral stability and self-adjointness under metric deformation.
• Analyzes the asymptotic expansion of the Green's function for the Dirac operator to define the mass endomorphism.
• Applies Sard's theorem and genericity arguments in Banach space settings to prove that zero sets of eigenspinors are empty for generic metrics.
• Uses local coordinate expansions and convergence estimates for metric-dependent endomorphisms and connection coefficients to control operator families.

Experimental results

Research questions

• RQ1 For which Riemannian metrics on a closed spin manifold of dimension 2 or 3 do non-harmonic eigenspinors have no zeros?
• RQ2 Is the mass endomorphism non-zero for a generic Riemannian metric on a closed spin manifold of dimension 3?
• RQ3 Can the existence of nowhere-vanishing eigenspinors be guaranteed generically in low-dimensional spin geometry?
• RQ4 How does the Dirac operator's spectral behavior depend on the Riemannian metric, particularly in terms of zero sets and conformal invariants?
• RQ5 What conditions ensure that the constant term in the Green's function expansion (the mass endomorphism) is non-zero?

Key findings

• On closed spin manifolds of dimension 2 or 3, non-harmonic eigenspinors have no zeros for a generic Riemannian metric.
• In dimension 3, the mass endomorphism is non-zero for a generic Riemannian metric, implying that the constant term in the Green's function expansion is generically non-vanishing.
• The Dirac operator's eigenspinors are generically nowhere vanishing under small perturbations of the metric, due to transversality and genericity theorems in Banach manifolds.
• The family of Dirac operators associated with a real-analytic one-parameter family of metrics extends to a self-adjoint holomorphic family of type (A), ensuring spectral regularity.
• The construction of the mass endomorphism via asymptotic expansion of the Green's function is well-defined and stable under generic metric variations.
• The results support the existence of nowhere-vanishing spinors, which are crucial for proving Hijazi's inequality and for constructing orthonormal frames in general relativity.