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Mathematische Zeitschrift

, Volume 284, Issue 1–2, pp 285–307 | Cite as

Existence of Dirac eigenvalues of higher multiplicity

  • Nikolai Nowaczyk
Article
  • 80 Downloads

Abstract

In this article, we prove that on any compact spin manifold of dimension \(m \equiv 0,6,7 \mod 8\), there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by “catching” the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.

Keywords

Spin geometry Dirac operator Spectral geometry Dirac spectrum Prescribing eigenvalues Surgery theory 

List of symbols

\({\bar{\beta }}_{g,h}\)

Identification isomorphism \(L^2(\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M) \rightarrow L^2(\Sigma ^h_{{{\mathrm{\mathbb {K}}}}} M)\)

\(D^l\)

Euclidean unit disc

\({\widehat{{{\mathrm{Diff}}}}}^{{{\mathrm{spin}}}}(M)\)

Group of spin diffeomorphisms with lift

\({{\mathrm{Diff}}}(M)\)

The diffeomorphism group of M

\({{\mathrm{Diff}}}^{{{\mathrm{spin}}}}(M)\)

group of spin diffeomorphisms

Open image in new window

Dirac operator w.r.t. g over \({{\mathrm{\mathbb {K}}}}\)

E(M)

A finite dimensional bundle over Y(M)

\(\Gamma (\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M)\)

Smooth spinor fields

\(H^1(\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M)\)

First order Sobolev space of sections of \(\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M\)

I

\(I := [0,1] \subset {{\mathrm{\mathbb {R}}}}\) is the unit interval

K

\(\mathbb {C}\) or \(\mathbb {H}\)

\({{\mathrm{\mathbb {K}}}}\)

\({{\mathrm{\mathbb {R}}}}\) or \(\mathbb {C}\)

\(L^2(\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M)\)

\(L^2\) spinor fields

\(L^2(\Sigma _{{{\mathrm{\mathbb {K}}}}} M)\)

Universal spinor field bundle

\(\lambda _j(g)\)

j-th eigenvalue of Open image in new window

M

A closed spin manifold of dimension m

\(\mu (\lambda )\)

Multiplicitiy of the eigenvalue \(\lambda \)

\(\mu _{{{\mathrm{\mathbb {K}}}}}(\lambda )\)

Multiplicitiy of the eigenvalue \(\lambda \) over \({{\mathrm{\mathbb {K}}}}\)

\({{\mathrm{\mathbb {N}}}}\)

The natural numbers \({{\mathrm{\mathbb {N}}}}=\{0,1,2, \ldots \}\)

\(\pi _{S^1}:I \rightarrow S^1\)

Canonical projection

\({{\mathrm{pr}}}^{{{\mathrm{spin}}}}\)

Projection \(\widehat{{{\mathrm{Diff}}}}^{{{\mathrm{spin}}}}(M) \rightarrow {{\mathrm{Diff}}}^{{{\mathrm{spin}}}}(M)\)

\(R_{\alpha }\)

Rotation by an angle \(\alpha \)

\({{\mathrm{\mathcal {R}}}}(M)\)

Space of Riemannian metrics on M with \(\mathcal {C}^1\)-topology

\(S^l\)

Euclidean unit sphere

\({{\mathrm{sgn}}}(E)\)

Sign of a vector bundle

\({{\mathrm{sgn}}}(f)\)

Sign of a loop

\(\Sigma ^g_{{{\mathrm{\mathbb {K}}}}} M\)

Spinor bundle over M w.r.t. g

Open image in new window

Dirac spectrum

\(\Theta \)

A topological spin structure

\(\mathcal {T}(M)\)

Smooth vector fields on M

X(M)

\(X(M) = {{\mathrm{\mathcal {R}}}}(M)\)

Y(M)

A subset of Riemannian metrics

\({{\mathrm{\mathbb {Z}}}}_2\)

\(\{\pm 1\}\)

Mathematics Subject Classification

53C27 58J05 58J50 57R65 

Notes

Acknowledgments

I would like to thank Bernd Ammann for interesting discussions. I’m also grateful to Mattias Dahl for explaining parts of his previous work to me. This research was kindly supported by the Studienstiftung des deutschen Volkes and the DFG Graduiertenkolleg GRK 1692 “Curvature, Cycles and Cohomology”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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