@phdthesis{Rothe2020,
  author    = {Rothe, Viktoria},
  title     = {Das Yamabe-Problem auf global-hyperbolischen Lorentz-Mannigfaltigkeiten},
  doi       = {10.25932/publishup-48601},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-486012},
  school      = {Universit{\"a}t Potsdam},
  pages     = {ix, 65},
  year      = {2020},
  abstract  = {Im Jahre 1960 behauptete Yamabe folgende Aussage bewiesen zu haben: Auf jeder kompakten Riemannschen Mannigfaltigkeit (M,g) der Dimension n ≥ 3 existiert eine zu g konform {\"a}quivalente Metrik mit konstanter Skalarkr{\"u}mmung. Diese Aussage ist {\"a}quivalent zur Existenz einer L{\"o}sung einer bestimmten semilinearen elliptischen Differentialgleichung, der Yamabe-Gleichung. 1968 fand Trudinger einen Fehler in seinem Beweis und infolgedessen besch{\"a}ftigten sich viele Mathematiker mit diesem nach Yamabe benannten Yamabe-Problem. In den 80er Jahren konnte durch die Arbeiten von Trudinger, Aubin und Schoen gezeigt werden, dass diese Aussage tats{\"a}chlich zutrifft. Dadurch ergeben sich viele Vorteile, z.B. kann beim Analysieren von konform invarianten partiellen Differentialgleichungen auf kompakten Riemannschen Mannigfaltigkeiten die Skalarkr{\"u}mmung als konstant vorausgesetzt werden. Es stellt sich nun die Frage, ob die entsprechende Aussage auch auf Lorentz-Mannigfaltigkeiten gilt. Das Lorentz'sche Yamabe Problem lautet somit: Existiert zu einer gegebenen r{\"a}umlich kompakten global-hyperbolischen Lorentz-Mannigfaltigkeit (M,g) eine zu g konform {\"a}quivalente Metrik mit konstanter Skalarkr{\"u}mmung? Das Ziel dieser Arbeit ist es, dieses Problem zu untersuchen. Bei der sich aus dieser Fragestellung ergebenden Yamabe-Gleichung handelt es sich um eine semilineare Wellengleichung, deren L{\"o}sung eine positive glatte Funktion ist und aus der sich der konforme Faktor ergibt. Um die f{\"u}r die Behandlung des Yamabe-Problems ben{\"o}tigten Grundlagen so allgemein wie m{\"o}glich zu halten, wird im ersten Teil dieser Arbeit die lokale Existenztheorie f{\"u}r beliebige semilineare Wellengleichungen f{\"u}r Schnitte auf Vektorb{\"u}ndeln im Rahmen eines Cauchy-Problems entwickelt. Hierzu wird der Umkehrsatz f{\"u}r Banachr{\"a}ume angewendet, um mithilfe von bereits existierenden Existenzergebnissen zu linearen Wellengleichungen, Existenzaussagen zu semilinearen Wellengleichungen machen zu k{\"o}nnen. Es wird bewiesen, dass, falls die Nichtlinearit{\"a}t bestimmte Bedingungen erf{\"u}llt, eine fast zeitglobale L{\"o}sung des Cauchy-Problems f{\"u}r kleine Anfangsdaten sowie eine zeitlokale L{\"o}sung f{\"u}r beliebige Anfangsdaten existiert. Der zweite Teil der Arbeit befasst sich mit der Yamabe-Gleichung auf global-hyperbolischen Lorentz-Mannigfaltigkeiten. Zuerst wird gezeigt, dass die Nichtlinearit{\"a}t der Yamabe-Gleichung die geforderten Bedingungen aus dem ersten Teil erf{\"u}llt, so dass, falls die Skalarkr{\"u}mmung der gegebenen Metrik nahe an einer Konstanten liegt, kleine Anfangsdaten existieren, so dass die Yamabe-Gleichung eine fast zeitglobale L{\"o}sung besitzt. Mithilfe von Energieabsch{\"a}tzungen wird anschließend f{\"u}r 4-dimensionale global-hyperbolische Lorentz-Mannigfaltigkeiten gezeigt, dass unter der Annahme, dass die konstante Skalarkr{\"u}mmung der konform {\"a}quivalenten Metrik nichtpositiv ist, eine zeitglobale L{\"o}sung der Yamabe-Gleichung existiert, die allerdings nicht notwendigerweise positiv ist. Außerdem wird gezeigt, dass, falls die H2-Norm der Skalarkr{\"u}mmung bez{\"u}glich der gegebenen Metrik auf einem kompakten Zeitintervall auf eine bestimmte Weise beschr{\"a}nkt ist, die L{\"o}sung positiv auf diesem Zeitintervall ist. Hierbei wird ebenfalls angenommen, dass die konstante Skalarkr{\"u}mmung der konform {\"a}quivalenten Metrik nichtpositiv ist. Falls zus{\"a}tzlich hierzu gilt, dass die Skalarkr{\"u}mmung bez{\"u}glich der gegebenen Metrik negativ ist und die Metrik gewisse Bedingungen erf{\"u}llt, dann ist die L{\"o}sung f{\"u}r alle Zeiten in einem kompakten Zeitintervall positiv, auf dem der Gradient der Skalarkr{\"u}mmung auf eine bestimmte Weise beschr{\"a}nkt ist. In beiden F{\"a}llen folgt unter den angef{\"u}hrten Bedingungen die Existenz einer zeitglobalen positiven L{\"o}sung, falls M = I x Σ f{\"u}r ein beschr{\"a}nktes offenes Intervall I ist. Zum Schluss wird f{\"u}r M = R x Σ ein Beispiel f{\"u}r die Nichtexistenz einer globalen positiven L{\"o}sung angef{\"u}hrt.},
  language  = {de}
}
@phdthesis{Hannes2022,
  author    = {Hannes, Sebastian},
  title     = {Boundary Value Problems for the Lorentzian Dirac Operator},
  doi       = {10.25932/publishup-54839},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-548391},
  school      = {Universit{\"a}t Potsdam},
  pages     = {67},
  year      = {2022},
  abstract  = {The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. B{\"a}r and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, B{\"a}r and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by B{\"a}r and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.},
  language  = {en}
}
@phdthesis{Gehring2023,
  author    = {Gehring, Penelope},
  title     = {Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary},
  doi       = {10.25932/publishup-57775},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577755},
  school      = {Universit{\"a}t Potsdam},
  pages     = {100},
  year      = {2023},
  abstract  = {Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, B{\"a}r and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While B{\"a}r and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed. We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems. Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions - the analogue of the Atiyah-Patodi-Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.},
  language  = {en}
}