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 is an ordered pair (*S*, [*h*]) where *S* is an oriented 2-manifold and [*h*] the set of all metrics conformally equivalent to a fixed 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 conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which are but not conformally diffeomorphic for any fixed . Finally, the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but not conformally diffeomorphic to any subset of the Minkowski 2-plane.

**[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:
http://dx.doi.org/10.1090/S0002-9939-1995-1273526-7

Article copyright:
© Copyright 1995
American Mathematical Society