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Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic


Authors: Robert W. Smyth and Tilla Weinstein
Journal: Proc. Amer. Math. Soc. 123 (1995), 3499-3506
MSC: Primary 53C50; Secondary 53A30
MathSciNet review: 1273526
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Abstract: A Lorentz surface $ \mathcal{L}$ is an ordered pair (S, [h]) where S is an oriented $ {C^\infty }$ 2-manifold and [h] the set of all $ {C^\infty }$ metrics conformally equivalent to a fixed $ {C^\infty }$ Lorentzian metric h on S. (Thus Lorentz surfaces are the indefinite metric analogs of Riemann surfaces.) This paper describes subsets of the Minkowski 2-plane which are conformally homeomorphic, but not even $ {C^1}$ conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which are $ {C^j}$ but not $ {C^{j + 1}}$ conformally diffeomorphic for any fixed $ j = 1,2, \ldots $. Finally, the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but not $ {C^1}$ conformally diffeomorphic to any subset of the Minkowski 2-plane.


References [Enhancements On Off] (What's this?)

  • [1] R. S. Kulkarni, An analogue of the Riemann mapping theorem for Lorentz metrics, Proc. Roy. Soc. London Ser. A 401 (1985), no. 1820, 117–130. MR 807317
  • [2] Tilla Weinstein, Inextendible conformal realizations of Lorentz surfaces in Minkowski 3-space, Michigan Math. J. 40 (1993), no. 3, 545–559. MR 1236178, 10.1307/mmj/1029004837
  • [3] -, An Introduction to Lorentz surfaces, Expositions in Math., de Gruyter, Berlin and New York (submitted).

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DOI: https://doi.org/10.1090/S0002-9939-1995-1273526-7
Article copyright: © Copyright 1995 American Mathematical Society