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Volume preserving curvature flows in Lorentzian manifolds

Makowski, Matthias

German Title: Volumenerhaltende Krümmungsflüsse in Lorentzmannigfaltigkeiten

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Abstract

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface and a curvature function F, which is either the mean curvature, the root of the second symmetric polynomial or a curvature function of class (K*). We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the graphs in the C-infinity topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.

Translation of abstract (German)

Sei N eine (n+1)-dimensionale global hyperbolische Lorentzmannigfaltigkeit mit einer kompakten Cauchy-Hyperflaeche und F eine Kruemmungsfunktion, welche entweder die mittlere Kruemmung, die Wurzel des zweiten elementarsymmetrischen Polynoms oder eine Kruemmungsfunktion der Klasse (K*) ist. Wir betrachten Kruemmungsfluesse mit Kruemmungsfunktion F und einem volumenerhaltendem Term und beweisen die Langzeitexistenz des Flusses, sowie die exponentielle Konvergenz der Graphen in der C-unendlich Topologie zu einer Hyperflaeche konstanter F-Kruemmung unter der Voraussetzung von Barrieren. Weiterhin untersuchen wir Stabilitaetseigenschaften und Blaetterungen von Hyperflaechen konstanter F-Kruemmung.

Item Type: Dissertation
Supervisor: Gerhardt, Prof. Dr. Claus
Date of thesis defense: 19 January 2011
Date Deposited: 02 Mar 2011 12:16
Date: 2010
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
Uncontrolled Keywords: Geometric analysis , Lorentz manifold , volume preserving , curvature flow
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