Geometric Regularization on Riemannian and Lorentzian Manifolds

  • Shantanu DaveEmail author
  • Günther Hörmann
  • Michael Kunzinger
Part of the Progress in Mathematics book series (PM, volume 301)


We investigate regularizations of distributional sections of vector bundles by means of nets of smooth sections that preserve the main regularity properties of the original distributions (singular support, wavefront set, Sobolev regularity). The underlying regularization mechanism is based on functional calculus of elliptic operators with finite speed of propagation with respect to a complete Riemannian metric. As an application we consider the interplay between the wave equation on a Lorentzian manifold and corresponding Riemannian regularizations, and under additional regularity assumptions we derive bounds on the rate of convergence of their commutator. We also show that the restriction to underlying space-like foliations behaves well with respect to these regularizations.


Regularization of distributions complete Riemannian manifolds globally hyperbolic Lorentz manifolds algebras of generalized functions 


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  1. 1.
    Ch. Bär, N. Ginoux, and F. Pfäffle. Wave equations on Lorentzian manifolds and quantization. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2007.Google Scholar
  2. 2.
    A.N. Bernal and M. Sánchez. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys., 257(1):43–50, 2005.zbMATHCrossRefGoogle Scholar
  3. 3.
    A.N. Bernal and M. Sánchez. Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’. Classical Quantum Gravity, 24(3):745–749, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J. Chazarain, A. Piriou. Introduction to the theory of linear partial differential equations. Studies in Mathematics and Its Applications, Vol. 14. North-Holland, 1982.Google Scholar
  5. 5.
    Y. Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009.Google Scholar
  6. 6.
    C.J.S. Clarke. The Analysis of Space-Time Singularities, Cambridge University Press, Cambridge, 1993.zbMATHGoogle Scholar
  7. 7.
    C.J.S. Clarke. Singularities: boundaries or internal points?, in Singularities, Black Holes and Cosmic Censorship, Joshi, P.S. and Raychaudhuri, A.K., eds., IUCCA, Bombay, 1996, pp. 24–32.Google Scholar
  8. 8.
    C.J.S. Clarke. Generalized hyperbolicity in singular spacetimes. Class. Quantum, Grav. 15 (1998), pp. 975–984.Google Scholar
  9. 9.
    J.F. Colombeau. New generalized functions and multiplication of distributions. North-Holland, Amsterdam, 1984.zbMATHGoogle Scholar
  10. 10.
    J.F. Colombeau. Elementary introduction to new generalized functions. North-Holland, Amsterdam, 1985.zbMATHGoogle Scholar
  11. 11.
    S. Dave. Geometrical embeddings of distributions into algebras of generalized functions, Math. Nachr., 283 (2010), no. 11, 1575–1588.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J.W. de Roever, M. Damsma. Colombeau algebras on a C -manifold. Indag. Math. (N.S.) 2 (1991), no. 3, 341–358.Google Scholar
  13. 13.
    S. Dave, G. Hörmann, and M. Kunzinger. Optimal regularization processes on complete Riemannian manifolds. 2010. arXiv:1003.3341 [math.FA].Google Scholar
  14. 14.
    N. Dapić, S. Pilipović, D. Scarpalezos. Microlocal analysis of Colombeau’s generalized functions: propagation of singularities. J. Anal. Math. 75 (1998), 51–66.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Dieudonné. Treatise on analysis. Vol. VIII, volume 10 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1993.Google Scholar
  16. 16.
    Garetto, C. Topological structures in Colombeau algebras: Topological ℂ˜-modules and duality theory. Acta Appl. Math. 88, No. 1, 81–123 (2005).Google Scholar
  17. 17.
    C. Garetto, G. Hörmann. Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc. (2) 48 (2005), no. 3, 603–629.Google Scholar
  18. 18.
    R. Geroch. Domain of dependence. Jour. Math. Phys., 11:437–449, 1970.MathSciNetzbMATHGoogle Scholar
  19. 19.
    J.D.E. Grant, E. Mayerhofer, R. Steinbauer. The wave equation on singular spacetimes. Comm. Math. Phys. 285 (2009), no. 2, 399–420.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer. Geometric theory of generalized functions, Kluwer, Dordrecht, 2001.Google Scholar
  21. 21.
    S. Haller. Microlocal analysis of generalized pullbacks of Colombeau functions. Acta Appl. Math. 105, no. 1, 83–109 (2009).Google Scholar
  22. 22.
    S.W. Hawking, G.F.R. Ellis. The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, 1973.Google Scholar
  23. 23.
    Higson N, Roe J (2000) Analytic 𝐾-homology. Oxford Mathematical Monographs, OxfordGoogle Scholar
  24. 24.
    G. H¨ormann. Integration and microlocal analysis in Colombeau algebras of generalized functions. J. Math. Anal. Appl. 239 (1999), no. 2, 332–348.Google Scholar
  25. 25.
    G. H¨ormann, M. Kunzinger, R. Steinbauer. Wave equations on non-smooth spacetimes, this volume, pp. 163–186.Google Scholar
  26. 26.
    A.A. Kosinski. Differential manifolds. Pure and Applied Mathematics 138, Academic Press (1993).Google Scholar
  27. 27.
    Kunzinger M, Steinbauer R (2002) Generalized pseudo-Riemannian geometry. Trans Amer Math Soc 354(10):4179–4199MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Leray J (1953) Hyperbolic Differential Equations. Lecture Notes, IAS, PrincetonGoogle Scholar
  29. 29.
    E. Minguzzi and M. S´anchez. The causal hierarchy of spacetimes. In Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., pages 299–358. Eur. Math. Soc., Z¨urich, 2008.Google Scholar
  30. 30.
    Nomizu K, Ozeki H (1961) The existence of complete Riemannian metrics. Proc Am Math Soc 12:889–891MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Oberguggenberger. Multiplication of distributions and applications to partial differential equations. Pitman Research Notes in Mathematics 59. Longman, New York, 1992.Google Scholar
  32. 32.
    O’Neill B (1983) Semi-Riemannian geometry, volume 103 of Pure and Applied Mathematics. Academic Press, New YorkGoogle Scholar
  33. 33.
    Steinbauer R, Vickers J (2006) The use of generalized functions and distributions in general relativity. Class Quantum Grav 23:R91–R114MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Vickers JA, Wilson JP (2000) Generalized hyperbolicity in conical spacetimes. Class Quantum Grav 17:1333–1260MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Shantanu Dave
    • 1
    Email author
  • Günther Hörmann
    • 1
  • Michael Kunzinger
    • 1
  1. 1.Faculty of MathematicsUniversity of ViennaWienAustria

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