# Continuity of Dirac spectra

@article{Nowaczyk2013ContinuityOD, title={Continuity of Dirac spectra}, author={Nikolai Nowaczyk}, journal={Annals of Global Analysis and Geometry}, year={2013}, volume={44}, pages={541-563} }

It is well known that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article, we extend this result to a global version. We view the spectrum of a Dirac operator as a function $$\mathbb Z \,\rightarrow \mathbb R \,$$Z→R and endow the space of all spectra with an $$\mathrm{arsinh }$$arsinh-uniform metric. We prove that the spectrum of the Dirac operator depends continuously…

## 9 Citations

### Existence of Dirac eigenvalues of higher multiplicity

- Mathematics
- 2016

In this article, we prove that on any compact spin manifold of dimension $$m \equiv 0,6,7 \mod 8$$m≡0,6,7mod8, there exists a metric, for which the associated Dirac operator has at least one…

### Dirac eigenvalues of higher multiplicity

- Mathematics
- 2015

Let M be a closed spin manifold of dimension at least three with a fixed topological spin structure. For any Riemannian metric, we can construct the associated Dirac operator. The spectrum of this…

### The Dirac operator under collapse to a smooth limit space

- MathematicsAnnals of Global Analysis and Geometry
- 2019

Let $$(M_i, g_i)_{i \in \mathbb {N}}$$ ( M i , g i ) i ∈ N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold ( B ,…

### Dirac operators with $W^{1,\infty}$-potential under codimension one collapse

- Mathematics
- 2017

We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1, \infty}$-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature…

### Dirac operators with $$W^{1,\infty }$$W1,∞-potential on collapsing sequences losing one dimension in the limit

- Mathematics
- 2018

We study the behavior of the spectrum of the Dirac operator together with a symmetric $$W^{1, \infty }$$W1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and…

### Dirac operators with W 1 , ∞ -potential on collapsing sequences losing one dimension in the limit

- Mathematics
- 2018

. We study the behavior of the spectrum of the Dirac operator together with a symmetric W 1 , ∞ -potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter…

### Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

- MathematicsCommunications in Mathematical Physics
- 2021

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum,…

### J ul 2 01 6 NON-NEGATIVE VERSUS POSITIVE SCALAR CURVATURE

- Mathematics
- 2018

: We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate…

## References

SHOWING 1-10 OF 15 REFERENCES

### Collapsing and Dirac-Type Operators

- Mathematics
- 2002

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the…

### Dirac eigenspinors for generic metrics

- Mathematics
- 2012

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…

### Generic Metrics and Connections on Spin- and Spinc-Manifolds

- Mathematics
- 1997

Abstract:We study the dependence of the dimension h0(g,A) of the kernel of the Atyiah-Singer Dirac operator
${\cal D}_{g,A}$ on a spinc-manifold M on the metric g and the connection A. The main…

### The Spectral Flow and the Maslov Index

- Mathematics
- 1995

exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of…

### Metrics with harmonic spinors

- Mathematics
- 1996

We show that every closed spin manifold of dimensionn ≡ 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a…

### Self-Adjoint Fredholm Operators And Spectral Flow

- MathematicsCanadian Mathematical Bulletin
- 1996

Abstract We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt } is a path of such operators, we can associate to…

### Generalized cylinders in semi-Riemannian and spin geometry

- Mathematics
- 2005

Abstract.We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result…

### Spineurs, opérateurs de dirac et variations de métriques

- Mathematics
- 1992

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie…

### The space of metrics of positive scalar curvature

- Mathematics
- 2014

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements in higher homotopy and homology groups…