Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic
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- by Robert W. Smyth and Tilla Weinstein PDF
- Proc. Amer. Math. Soc. 123 (1995), 3499-3506 Request permission
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
- 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
- Tilla Weinstein, Inextendible conformal realizations of Lorentz surfaces in Minkowski $3$-space, Michigan Math. J. 40 (1993), no. 3, 545–559. MR 1236178, DOI 10.1307/mmj/1029004837 —, An Introduction to Lorentz surfaces, Expositions in Math., de Gruyter, Berlin and New York (submitted).
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3499-3506
- MSC: Primary 53C50; Secondary 53A30
- DOI: https://doi.org/10.1090/S0002-9939-1995-1273526-7
- MathSciNet review: 1273526