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The Ricci Flow of Asymptotically Hyperbolic Mass Open Access


Other title
Positive Mass
Horowitz and Myers Conjecture
Parabolic PDE
Hyperbolic Space
Ricci Flow
Asymptotically Hyperbolic
Type of item
Degree grantor
University of Alberta
Author or creator
Balehowsky, Tracey J
Supervisor and department
Woolgar, Eric (Mathematical and Statistical Sciences)
Examining committee member and department
Kuttler, Jochen (Mathematical and Statistical Sciences)
Morsink, Sharon (Physics)
van Roessel, Henry (Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Master of Science
Degree level
In this thesis, we generalize the notion of asymptotically hyperbolic mass (first introduced by Wang in 2001) to manifolds with toroidal ends. Using this generalized definition, we show that under a normalized Ricci flow with asymptotically hyperbolic, conformally compact initial data with a well-defined mass, the mass will decay exponentially in time to zero, in contradistinction to the constant behaviour of asymptotically flat mass under Ricci flow. We then use this result for the evolution of asymptotically hyperbolic mass to prove that there does not exist a breather solution to the normalized Ricci flow with non-zero mass. Further, we provide a proof of the rigidity case of the Positive Mass Theorem in the asymptotically hyperbolic setting, using Ricci flow. We note that this result for the exponential behaviour of asymptotically hyperbolic mass provides support for a conjecture in general relativity stated by Horowitz and Myers.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
T Balehowsky, E Woolgar, The Ricci Flow of Asymptotically Hyperbolic Mass and Applications, (2012) Journal of Mathematical Physics, 53: 072501.

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