Mass endomorphism, surgery and perturbations

Abstract. The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a generalized Killing spinor on M . We show, conversely, that in the real analytic category, every spin manifold (M, g) carrying a generalized Killing spinor ψ can be isometrically embedded as a hypersurface in a spin manifold carrying a parallel spinor whose restriction to M is ψ. We also answer negatively the corresponding question in the smooth category.

show abstract

“…22 ([33] for n = 3,[4] for n ≥ 3). For generic metrics in R U,g flat (M) the mass endomorphism in p is non-zero.…”

mentioning

confidence: 99%

The Cauchy Problems for Einstein Metrics and Parallel Spinors

Ammann

Moroianu

Moroianu³

2013

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A series of works of B. Ammann and his group [3][4][5][6][7][8][9] have provided a brief picture of how variational method is employed to the study of (1.1). From the view point of analysis, as it was pointed out in [3], standard variational methods do not directly imply the existence of a solution.…”

Section: Introductionmentioning

confidence: 99%

Curvature effect in the spinorial Yamabe problem on product manifolds

Bartsch,

2023

Preprint

View full text Add to dashboard Cite

Let (M 1 , g (1) ), (M 2 , g (2) ) be closed Riemannian spin manifolds. We study the existence of solutions of the Spinorial Yamabe problem on the product M 1 × M 2 equipped with a family of metrics ε −2 g (1) ⊕ g (2) , ε > 0. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor M 1 , and which exhibit a spike layer as ε → 0. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on (M 1 , g (1) ).

show abstract

“…In order to describe the dependence of the mass endomorphism on the Riemannian metrics, let M U,flat (M) be the set of all Riemannian metrics g on M, which are flat on an open subset U M, and M inv U,flat (M) ⊂ M U,flat (M) the subset of metrics with invertible Dirac operators. Then it was proven in [8], see also [21], that for dimension m ≥ 3, the subset…”

Section: Introductionmentioning

confidence: 99%

On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem

Sire¹,

Xu²

2021

Preprint

View full text Add to dashboard Cite

Let M be a closed spin manifold of dimension m ≥ 6 equipped with a Riemannian metric g and a spin structure σ. Let λ + 1 (g) be the smallest positive eigenvalue of the Dirac operator D g on M with respect to a metric g conformal to g. The Bär-Hijazi-Lott invariant is defined by λ. In this paper, we show that1 m provided that g is not locally conformally flat. This estimate is a spinorial analogue to an estimate by T. Aubin, solving the Yamabe problem in this setting.

show abstract

Mass endomorphism, surgery and perturbations

Abstract: We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments

Cited by 7 publications

References 19 publications

The Cauchy Problems for Einstein Metrics and Parallel Spinors

The Cauchy Problems for Einstein Metrics and Parallel Spinors

Curvature effect in the spinorial Yamabe problem on product manifolds

On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem

Contact Info

Product

Resources

About