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  • 1.
    Ames, Ellery
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Beyer, F.
    Isenberg, J.
    LeFloch, P. G.
    A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges2017In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 121, p. 42-71Article in journal (Refereed)
    Abstract [en]

    We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave gauges. These results show that the AVTD property, which has so far been known to hold for solutions in areal coordinates only, is stable to perturbations of the coordinate systems. Our proof is based on an analysis of the singular initial value problem for the Einstein vacuum equations in the generalized wave gauge formalism, and provides a framework which we anticipate to be useful for more general spacetimes.

  • 2. Cabrera Pacheco, A. J.
    et al.
    Cederbaum, C.
    McCormick, Stephen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Asymptotically hyperbolic extensions and an analogue of the Bartnik mass2018In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 132, p. 338-357Article in journal (Refereed)
    Abstract [en]

    The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen (2016), of Miao and Xie (2018), and of joint work of Miao and the authors (Cabrera Pacheco et al., 2017), we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature. 

  • 3. Carey, Alan L.
    et al.
    Mickelsson, Jouko
    KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
    Wang, Bai-Ling
    Differential twisted K-theory and applications2009In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 59, no 5, p. 632-653Article in journal (Refereed)
    Abstract [en]

    In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

  • 4.
    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).

  • 5. Escher, J.
    et al.
    Kohlmann, M.
    Lenells, Jonatan
    Department of Mathematics, Baylor University, , Waco, United States .
    The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations2011In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 61, no 2, p. 436-452Article in journal (Refereed)
    Abstract [en]

    We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S1) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature. © 2010 Elsevier B.V.

  • 6.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics. KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Descent equations of Yang-Mills anomalies in noncommutative geometry1997In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, p. 259-279Article in journal (Refereed)
  • 7.
    Lee, Jae Min
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Geometric approach on the global conservative solutions of the Camassa–Holm equation2019In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 142, p. 137-150Article in journal (Refereed)
    Abstract [en]

    In this paper we construct global weak conservative solutions of the Camassa–Holm (CH)equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using the change of variables ρ=η x . The new variable removes the singularity of the CH equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter–Saxton equation using the geometric interpretation.

  • 8.
    Lenells, Jonatan
    University of California, United States .
    The Hunter-Saxton equation describes the geodesic flow on a sphere2007In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 57, no 10, p. 2049-2064Article in journal (Refereed)
    Abstract [en]

    The Hunter-Saxton equation is the Euler equation for the geodesic flow on the quotient space Rot (S) {set minus} D (S) of the infinite-dimensional group D (S) of orientation-preserving diffeomorphisms of the unit circle S modulo the subgroup of rotations Rot (S) equipped with the over(H, ̇)1 right-invariant metric. We establish several properties of the Riemannian manifold Rot (S) {set minus} D (S): it has constant curvature equal to 1, the Riemannian exponential map provides global normal coordinates, and there exists a unique length-minimizing geodesic joining any two points of the space. Moreover, we give explicit formulas for the Jacobi fields, we prove that the diameter of the manifold is exactly frac(π, 2), and we give exact estimates for how fast the geodesics spread apart. At the end, these results are given a geometric and intuitive understanding when an isometry from Rot (S) {set minus} D (S) to an open subset of an L2-sphere is constructed.

  • 9.
    Lenells, Jonatan
    et al.
    Baylor University, United States .
    Yang, Z.
    A two-component geodesic equation on a space of constant positive curvature2012In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 62, no 5, p. 1298-1308Article in journal (Refereed)
    Abstract [en]

    We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.

  • 10.
    Mickelsson, Jouko
    et al.
    KTH, School of Engineering Sciences (SCI), Theoretical Physics.
    Paycha, Sylvie
    Renormalised Chern-Weil forms associated with families of Dirac operators2007In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 57, no 9, p. 1789-1814Article in journal (Refereed)
    Abstract [en]

    We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [S. Scott, Zeta-Chern forms and the local family index theorem, Trans. Amer. Math. Soc. (in press). arXiv: math.DG/0406294] and later in [S. Paycha, S. Scott, Chern-Weil forms associated with superconnections, in: B. Booss-Bavnbeck, S. Klimek, M. Lesch, W. Zhang (Eds.), Analysis, Geometry and Topology of Elliptic Operators, World Scientific, 2006]. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern-Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for eta-forms.

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