Uncountably many conformally distinct Lorentz surfaces and a finiteness theorem

Author:
Robert W. Smyth

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1559-1566

MSC (1991):
Primary 53C50, 53A30

DOI:
https://doi.org/10.1090/S0002-9939-96-03558-7

MathSciNet review:
1346988

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Abstract: This paper describes an uncountable family of Lorentz surfaces realized as rectangular regions in the Minkowski 2-plane . A simple conformal invariant is defined which assigns a different real value to each Lorentz surface in the family. While these surfaces provide uncountably many conformally distinct, bounded, convex subsets of which are each symmetric about a properly embedded timelike curve and about a properly embedded spacelike curve, it is shown that there are only 21 conformally distinct, bounded, convex subsets of which are symmetric about some null line.

**1.**R. Kulkarni,*An analogue of the Riemann mapping theorem for Lorentz metrics*, Proc. R. Soc. Lond. A**401**(1985), 117-130. MR**87e:53108****2.**F. Luo and R. Stong,*An analogue of the Riemann mapping theorem for Lorentz metrics: Topological Embedding of a Twice Foliated Disc into the Plane*, preprint.**3.**R. Smyth and T. Weinstein,*Conformally Homeomorphic Lorentz Surfaces Need Not Be Conformally Diffeomorphic*, Proc. Amer. Math. Soc.**123**(1995), 3499--3506. CMP**95:16****4.**T. Weinstein,*An Introduction to Lorentz Surfaces*, DeGruyter Expositions in Math. (to appear).

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Additional Information

**Robert W. Smyth**

Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Address at time of publication:
Department of Mathematics, Georgian Court College, Lakewood, New Jersey 08701

Email:
rsmyth@math.rutgers.edu, rsmyth@georgian.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03558-7

Keywords:
Indefinite metric,
conformal geometry

Received by editor(s):
October 20, 1994

Communicated by:
Christopher Croke

Article copyright:
© Copyright 1996
American Mathematical Society