The Dirac operator under collapse with bounded curvature and diameter
The Dirac operator under collapse with bounded curvature and diameter
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DissertationPrüfungsdatum
06.09.2018Datum der Veröffentlichung
26.09.2018Erstgutachter
Ballmann, WernerZweitgutachter
Ammann, BerndMetadaten
Zur Langanzeige
Zitierbare Links 
Inhalt
A sequence (Mi, gi)i of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (B,h). The limit space (B,h) has in general many singularities.
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifoldsi divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold.
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifoldsi divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold.
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.
Schlagwörter
Spin-Strukturen, Eigenwert, Grenzberechnung, Faserbündel, Nilpotente Lie-Gruppe, Spin structure, eigenvalue, calculation of limits, fiber bundles, nilpotent lie group
Klassifikation (DDC)
510 MathematikRoos, Saskia Christine: The Dirac operator under collapse with bounded curvature and diameter. - Bonn, 2018. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-51961
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-51961
@phdthesis{handle:20.500.11811/7638,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-51961,
author = {{Saskia Christine Roos}},
title = {The Dirac operator under collapse with bounded curvature and diameter},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2018,
month = sep,
note = {A sequence (Mi, gi)i of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (B,h). The limit space (B,h) has in general many singularities.
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifoldsi divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold.
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.},
url = {https://hdl.handle.net/20.500.11811/7638}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-51961,
author = {{Saskia Christine Roos}},
title = {The Dirac operator under collapse with bounded curvature and diameter},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2018,
month = sep,
note = {A sequence (Mi, gi)i of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (B,h). The limit space (B,h) has in general many singularities.
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifoldsi divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold.
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.},
url = {https://hdl.handle.net/20.500.11811/7638}
}
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