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A study of asymptotically hyperbolic manifolds in mathematical relativity
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of ve papers where certain problems arising in mathematical relativity are studied in the context of asymptotically hyperbolic manifolds.

In Paper A we deal with constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds. Conditions on the scalar field and its potential are given which lead to existence and non-existence results.

In Paper B we construct non-constant mean curvature solutions of the constraint equations on asymptotically hyperbolic manifolds. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then letting the exponent tend to its true value. We prove that if a certain limit equation admits no non-trivial solution, then the set of solutions of the constraint equations is non empty and compact. W ealso give conditions ensuring that the limit equation admits no nontrivial solution. This is a joint work with Romain Gicquaud.

In this Paper C we obtain Penrose type inequalities for asymptotically hyperbolic graphs. In certain cases we prove that equality is attained only by the anti-de Sitter Schwarzschild metric. This is a joint work with Mattias Dahl and Romain Gicquaud.

In Paper D we construct a solution to the Jang equation on an asymptotically hyperbolic manifold with a certain asymptotic behaviour at infinity.

In Paper E we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that when the mass tends to zero the metric converges uniformly tot he hyperbolic metric. This is a joint work with Mattias Dahl and Romain Gicquaud.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. , vii, 52 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 12:05
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-102874OAI: oai:DiVA.org:kth-102874DiVA: diva2:557156
Public defence
2012-10-15, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20120928

Available from: 2012-09-28 Created: 2012-09-27 Last updated: 2012-09-28Bibliographically approved
List of papers
1. Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds
Open this publication in new window or tab >>Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds
2010 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, no 24, 245019- p.Article in journal (Refereed) Published
Keyword
Einstein-scalar field equations, constraint equations, asymptotically hyperbolic manifold, conformal method, constant mean curvature
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-27201 (URN)10.1088/0264-9381/27/24/245019 (DOI)000284829400019 ()
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20101209 Uppdaterad från manuskript till published 20101221

Available from: 2010-12-09 Created: 2010-12-09 Last updated: 2017-12-11Bibliographically approved
2. A Large Class of Non-Constant Mean Curvature Solutions of the Einstein Constraint Equations on an Asymptotically Hyperbolic Manifold
Open this publication in new window or tab >>A Large Class of Non-Constant Mean Curvature Solutions of the Einstein Constraint Equations on an Asymptotically Hyperbolic Manifold
2012 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 310, no 3, 705-763 p.Article in journal (Refereed) Published
Abstract [en]

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non-trivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-27202 (URN)10.1007/s00220-012-1420-4 (DOI)000301492300006 ()
Note

QC 20120411

Available from: 2010-12-09 Created: 2010-12-09 Last updated: 2017-12-11Bibliographically approved
3. Penrose type inequalities for asymptotically hyperbolic graphs
Open this publication in new window or tab >>Penrose type inequalities for asymptotically hyperbolic graphs
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-102868 (URN)
Note

QS 2012

Available from: 2012-09-27 Created: 2012-09-27 Last updated: 2012-09-28Bibliographically approved
4. The Jang equation on an asymptotically hyperbolic manifold
Open this publication in new window or tab >>The Jang equation on an asymptotically hyperbolic manifold
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-102871 (URN)
Note

QS 2012

Available from: 2012-09-27 Created: 2012-09-27 Last updated: 2012-09-28Bibliographically approved
5. Asymptotically Hyperbolic Manifolds with Small Mass
Open this publication in new window or tab >>Asymptotically Hyperbolic Manifolds with Small Mass
2014 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 325, no 2, 757-801 p.Article in journal (Refereed) Published
Abstract [en]

For asymptotically hyperbolic manifolds of dimension n with scalar curvature at least equal to -n(n - 1) the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-102872 (URN)10.1007/s00220-013-1827-6 (DOI)000329583800009 ()2-s2.0-84891903390 (Scopus ID)
Note

QC 20140205. Updated from manuscript to article in journal.

Available from: 2012-09-27 Created: 2012-09-27 Last updated: 2017-12-07Bibliographically approved

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