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  • 1. Ammann, B.
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, E.
    The conformal Yamabe constant of product manifolds2013In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 141, no 1, p. 295-307Article in journal (Refereed)
    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).

  • 2. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hermann, Andreas
    Humbert, Emmanuel
    Mass endomorphism, surgery and perturbations2014In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 64, no 2, p. 467-487Article in journal (Refereed)
    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.

  • 3. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Harmonic spinors and local deformations of the metric2011In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 18, no 5, p. 927-936Article in journal (Refereed)
    Abstract [en]

    Let (M, g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.

  • 4. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Low-dimensional surgery and the Yamabe invariant2015In: Journal of the Mathematical Society of Japan, ISSN 0025-5645, E-ISSN 1881-1167, Vol. 67, no 1, p. 159-182Article in journal (Refereed)
    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.

  • 5. Ammann, Bernd
    et al.
    Dahl, Mattias
    Humbert, Emmanuel
    Smooth yamabe invariant and surgery2013In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 94, no 1, p. 1-58Article in journal (Refereed)
    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)}.

  • 6. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Square-integrability of solutions of the Yamabe equation2013In: Communications in analysis and geometry, ISSN 1019-8385, E-ISSN 1944-9992, Vol. 21, no 5, p. 891-916Article in journal (Refereed)
    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.

  • 7. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Surgery and harmonic spinors2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 220, no 2, p. 523-539Article in journal (Refereed)
    Abstract [en]

    Let M he a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.

  • 8. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Surgery and the Spinorial tau-Invariant2009In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 34, no 10, p. 1147-1179Article in journal (Refereed)
    Abstract [en]

    We associate to a compact spin manifold M a real-valued invariant (M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's sigma-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then (N) epsilon min{(M), n}, where n is a positive constant depending only on n=dim M. Various topological conclusions can be drawn, in particular that is a spin-bordism invariant below n. Also, below n the values of cannot accumulate from above when varied over all manifolds of dimension n.

  • 9. Andersson, L.
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Galloway, G. J.
    Pollack, D.
    On the geometry and topology of initial data sets with horizons2018In: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 22, no 5, p. 863-882Article in journal (Refereed)
    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.

  • 10. Andersson, Lars
    et al.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    scalar curvature rigidity for asymptotically locally hyperbolic manifolds1998In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 16, p. 1-27Article in journal (Refereed)
    Abstract [en]

    Rigidity results for asymptotically locally hyperbolic manifoldswith lower bounds on scalar curvature are proved using spinor methodsrelated to the Witten proof of the positive mass theorem. The argument isbased on a study of the Dirac operator defined with respect to the Killingconnection. The existence of asymptotic Killing spinors is related to thespin structure on the end. The expression for the mass is calculated andproven to vanish for conformally compact Einstein manifolds with conformalboundary a spherical space form, giving rigidity. In the 4-dimensional case,the signature of the manifold is related to the spin structure on the end andexplicit formulas for the relevant invariants are given.

  • 11.
    Andersson, Lars
    et al.
    KTH, Superseded Departments, Mathematics.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Howard, Ralph
    Boundary and lens rigidity of Lorentzian surfaces1996In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 348, p. 2307-2329Article in journal (Refereed)
    Abstract [en]

    Let g be a Lorentzian metric on the plane ℝ2 that agrees with the standard metric g0 = -dx2 + dy2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (ℝ2, g). Then (ℝ2, g) and (ℝ2, g0) are isometric. Further, if (M*, g*) and (M*, p*) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesies of (M, g) maximize the distances between their points and (M, g) and (M*, g*) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M*, g*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

  • 12. Bär, C.
    et al.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    Small eigenvalues of the Conformal Laplacian2003In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 13, no 3, p. 483-508Article in journal (Refereed)
    Abstract [en]

    We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the alpha-genus.

  • 13. Bär, C.
    et al.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    The first Dirac eigenvalues on manifolds with positive scalar curvature2004In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 132, no 11, p. 3337-3344Article in journal (Refereed)
    Abstract [en]

    We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

  • 14. Bär, Christian
    et al.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    Surgery and the spectrum of the Dirac operator2002In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 552, p. 53-76Article in journal (Refereed)
    Abstract [en]

    We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension greater than or equal to5 the dimension of the space of harmonic spinors is not larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension greater than or equal to3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.

  • 15.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    Dependence on the spin structure of the eta and Rokhlin invariants2002In: Topology and its Applications, ISSN 0166-8641, E-ISSN 1879-3207, Vol. 118, no 3, p. 345-355Article in journal (Refereed)
    Abstract [en]

    We study the dependence of the eta invariant etaD on the spin structure, where D is a twisted Dirac operator on a (4k + 3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a cohomology class which is the reduction of a H-1 (M, Z)-class is shown to be a half integer. As an application of the technique of proof the generalized Rokhlin invariant is shown to be equal modulo 8 for two spin structures related in this way.

  • 16.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Dirac eigenvalues for generic metrics on three-manifolds2003In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 24, p. 95-100Article in journal (Refereed)
    Abstract [en]

    We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.

  • 17.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the space of metrics with invertible Dirac operator2008In: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 83, p. 451-469Article in journal (Refereed)
    Abstract [en]

    On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that, if non-empty, the space of metrics with invertible Dirac operators is disconnected in dimensions n equivalent to 0, 1, 3, 7 mod 8, n >= 5. As corollaries follow results on the existence of metrics with harmonic spinors by Hitchin and Bar. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics with harmonic spinors on simply connected manifolds with a cyclic group action. In particular this applies to spheres of all dimensions n >= 5.

  • 18.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Prescribing eigenvalues of the Dirac operator2005In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 118, no 2, p. 191-199Article in journal (Refereed)
    Abstract [en]

    In this note we show that every compact spin manifold of dimension <= 3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.

  • 19.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    Some applications of dirac operators and eta invariants in geometry1999Doctoral thesis, comprehensive summary (Other scientific)
  • 20.
    Dahl, Mattias
    KTH, Superseded Departments, Mathematics.
    The Positive Mass Theorem for Ale Manifolds1997In: Mathematics of gravitation. P. 1: Lorentzian geometry and Einstein equations / [ed] Piotr T. Chruściel, 1997Conference paper (Other academic)
  • 21.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gicquaud, Romain
    Humbert, Emmanuel
    A Limit Equation Associated To The Solvability Of The Vacuum Einstein Constraint Equations By Using The Conformal Method2012In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 161, no 14, p. 2669-2697Article in journal (Refereed)
    Abstract [en]

    Let (M, g) be a compact Riemannian manifold on which a trace-free and divergence-free sigma is an element of W-1,W-p and a positive function tau is an element of W-1,W-p, p > n are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data sigma and tau. We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on 1-forms. This last equation can be shown to be without solutions in many situations. As a corollary, we get the existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which, in particular, hold on a dense set of metrics g for the C-0-topology.

  • 22.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gicquaud, Romain
    Humbert, Emmanuel
    A non-existence result for a generalization of the equations of the conformal method in general relativity2013In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 30, no 7, p. 075004-Article in journal (Refereed)
    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.

  • 23.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gicquaud, Romain
    Sakovich, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Asymptotically Hyperbolic Manifolds with Small Mass2014In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 325, no 2, p. 757-801Article in journal (Refereed)
    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.

  • 24.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gicquaud, Romain
    Sakovich, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Penrose type inequalities for asymptotically hyperbolic graphsManuscript (preprint) (Other academic)
  • 25.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Gicquaud, Romain
    Sakovich, Anna
    Penrose Type Inequalities for Asymptotically Hyperbolic Graphs2013In: 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)
    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.

  • 26.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Grosse, Nadine
    Invertible Dirac operators and handle attachments on manifolds with boundary2014In: Journal of Topology and Analysis (JTA), ISSN 1793-5253, E-ISSN 1793-7167, Vol. 6, no 3, p. 339-382Article in journal (Refereed)
    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.

  • 27.
    Dahl, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    An isoperimetric constant associated to horizons in S(3) blown up at two points2011In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 61, no 10, p. 1809-1822Article in journal (Refereed)
    Abstract [en]

    Let g be a metric on S(3) with positive Yamabe constant. When blowing up g at two points, a scalar flat manifold with two asymptotically flat ends is produced and this manifold will have compact minimal surfaces. We introduce the Theta-invariant for g which is an isoperimetric constant for the cylindrical domain inside the outermost minimal surface of the blown-up metric. Further we find relations between Theta and the Yamabe constant and the existence of horizons in the blown-up metric on R(3).

1 - 27 of 27
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