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Publications (10 of 27) Show all publications
Andersson, L., Dahl, M., Galloway, G. J. & Pollack, D. (2018). On the geometry and topology of initial data sets with horizons. Asian Journal of Mathematics, 22(5), 863-882
Open this publication in new window or tab >>On the geometry and topology of initial data sets with horizons
2018 (English)In: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 22, no 5, p. 863-882Article in journal (Refereed) Published
Abstract [en]

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M, g,K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M \ ∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension n = 3, to higher dimensions.

Place, publisher, year, edition, pages
International Press of Boston, Inc., 2018
Keywords
Initial data set, Jang's equation, Marginally outer trapped surfaces
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-246525 (URN)10.4310/AJM.2018.v22.n5.a4 (DOI)2-s2.0-85058986797 (Scopus ID)
Note

QC 20190320

Available from: 2019-03-19 Created: 2019-03-20 Last updated: 2019-03-20Bibliographically approved
Ammann, B., Dahl, M. & Humbert, E. (2015). Low-dimensional surgery and the Yamabe invariant. Journal of the Mathematical Society of Japan, 67(1), 159-182
Open this publication in new window or tab >>Low-dimensional surgery and the Yamabe invariant
2015 (English)In: Journal of the Mathematical Society of Japan, ISSN 0025-5645, E-ISSN 1881-1167, Vol. 67, no 1, p. 159-182Article in journal (Refereed) Published
Abstract [en]

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k <= n - 3. The smooth Yamabe invariants sigma(M) and sigma(N) satisfy sigma(N) >= min(sigma(M), Lambda) for a constant Lambda > 0 depending only on n and k. We derive explicit positive lower bounds for A in dimensions where previous methods failed, namely for (n, k) is an element of {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

Keywords
Yamabe invariant, surgery, symmetrization
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-161148 (URN)10.2969/jmsj/06710159 (DOI)000348690300005 ()2-s2.0-84921881989 (Scopus ID)
Note

QC 20150319

Available from: 2015-03-19 Created: 2015-03-09 Last updated: 2017-12-04Bibliographically approved
Dahl, M., Gicquaud, R. & Sakovich, A. (2014). Asymptotically Hyperbolic Manifolds with Small Mass. Communications in Mathematical Physics, 325(2), 757-801
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, p. 757-801Article 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
Dahl, M. & Grosse, N. (2014). Invertible Dirac operators and handle attachments on manifolds with boundary. Journal of Topology and Analysis (JTA), 6(3), 339-382
Open this publication in new window or tab >>Invertible Dirac operators and handle attachments on manifolds with boundary
2014 (English)In: Journal of Topology and Analysis (JTA), ISSN 1793-5253, E-ISSN 1793-7167, Vol. 6, no 3, p. 339-382Article in journal (Refereed) Published
Abstract [en]

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.

Keywords
Spectrum of the Dirac operator, manifold with boundary, handle attachment, concordance of Riemannian metrics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-148336 (URN)10.1142/S1793525314500137 (DOI)000337891400002 ()2-s2.0-84902458207 (Scopus ID)
Note

QC 20140806

Available from: 2014-08-06 Created: 2014-08-05 Last updated: 2017-12-05Bibliographically approved
Ammann, B., Dahl, M., Hermann, A. & Humbert, E. (2014). Mass endomorphism, surgery and perturbations. Annales de l'Institut Fourier, 64(2), 467-487
Open this publication in new window or tab >>Mass endomorphism, surgery and perturbations
2014 (English)In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 64, no 2, p. 467-487Article in journal (Refereed) Published
Abstract [en]

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

Keywords
Dirac operator, mass endomorphism, surgery
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-160006 (URN)000348011300004 ()2-s2.0-84919400502 (Scopus ID)
Note

QC 20150218

Available from: 2015-02-18 Created: 2015-02-12 Last updated: 2017-12-04Bibliographically approved
Dahl, M., Gicquaud, R. & Humbert, E. (2013). A non-existence result for a generalization of the equations of the conformal method in general relativity. Classical and quantum gravity, 30(7), 075004
Open this publication in new window or tab >>A non-existence result for a generalization of the equations of the conformal method in general relativity
2013 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 30, no 7, p. 075004-Article in journal (Refereed) Published
Abstract [en]

The constraint equations of general relativity can in many cases be solved by the conformal method. We show that a slight modification of the equations of the conformal method admits no solution for a broad range of parameters. This suggests that the question of existence or non-existence of solutions to the original equations is more subtle than could perhaps be expected.

Keywords
Einstein Constraint Equations, Mean-Curvature
National Category
Physical Sciences Mathematics
Identifiers
urn:nbn:se:kth:diva-120523 (URN)10.1088/0264-9381/30/7/075004 (DOI)000316227500004 ()2-s2.0-84875239988 (Scopus ID)
Note

QC 20130411

Available from: 2013-04-11 Created: 2013-04-11 Last updated: 2017-12-06Bibliographically approved
Dahl, M., Gicquaud, R. & Sakovich, A. (2013). Penrose Type Inequalities for Asymptotically Hyperbolic Graphs. Annales de l'Institute Henri Poincare. Physique theorique, 14(5), 1135-1168
Open this publication in new window or tab >>Penrose Type Inequalities for Asymptotically Hyperbolic Graphs
2013 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 14, no 5, p. 1135-1168Article in journal (Refereed) Published
Abstract [en]

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space H-n. The graphs are considered as unbounded hypersurfaces of Hn+1 which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-124567 (URN)10.1007/s00023-012-0218-4 (DOI)000320002900004 ()2-s2.0-84879988595 (Scopus ID)
Note

QC 20130712

Available from: 2013-07-12 Created: 2013-07-12 Last updated: 2017-12-06Bibliographically approved
Ammann, B., Dahl, M. & Humbert, E. (2013). Smooth yamabe invariant and surgery. Journal of differential geometry, 94(1), 1-58
Open this publication in new window or tab >>Smooth yamabe invariant and surgery
2013 (English)In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 94, no 1, p. 1-58Article in journal (Refereed) Published
Abstract [en]

We prove a surgery formula for the smooth Yamabe invariant sigma(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Lambda(n), depending only on the dimension n of M, such that sigma(N) >= min{sigma(M), Lambda(n)}.

National Category
Geometry
Identifiers
urn:nbn:se:kth:diva-124467 (URN)000319243900001 ()2-s2.0-84879936537 (Scopus ID)
Note

QC 20130709

Available from: 2013-07-09 Created: 2013-07-05 Last updated: 2017-12-06Bibliographically approved
Ammann, B., Dahl, M. & Humbert, E. (2013). Square-integrability of solutions of the Yamabe equation. Communications in analysis and geometry, 21(5), 891-916
Open this publication in new window or tab >>Square-integrability of solutions of the Yamabe equation
2013 (English)In: Communications in analysis and geometry, ISSN 1019-8385, E-ISSN 1944-9992, Vol. 21, no 5, p. 891-916Article in journal (Refereed) Published
Abstract [en]

We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds, which are bounded and L-p for p = 2n/(n -2) are also L-2. This L-p-L-2 implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in our paper [4]. As an application we see that the smooth Yamabe invariant of any two-connected compact seven-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions >= 11.

Keywords
Scalar Curvature, Spin Cobordism, Manifolds, Invariant, Surgery
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-140381 (URN)10.4310/CAG.2013.v21.n5.a2 (DOI)000328958800002 ()2-s2.0-84891876569 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20140123

Available from: 2014-01-23 Created: 2014-01-23 Last updated: 2017-12-06Bibliographically approved
Ammann, B., Dahl, M. & Humbert, E. (2013). The conformal Yamabe constant of product manifolds. Proceedings of the American Mathematical Society, 141(1), 295-307
Open this publication in new window or tab >>The conformal Yamabe constant of product manifolds
2013 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 141, no 1, p. 295-307Article in journal (Refereed) Published
Abstract [en]

Let (V, g) and (W, h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V × W, g + h) in terms of the conformal Yamabe constants of (V, g) and (W, h).

Keywords
Product manifolds, Yamabe constant, Yamabe invariant
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-106172 (URN)10.1090/S0002-9939-2012-11320-6 (DOI)000326513700026 ()2-s2.0-84868119184 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20121129

Available from: 2012-11-29 Created: 2012-11-29 Last updated: 2017-12-07Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-9184-1467

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