|
Regularity of Lorentzian Busemann Functions
Author(s):
Gregory
J.
Galloway;
Arnaldo
Horta
Journal:
Trans. Amer. Math. Soc.
348
(1996),
2063-2084.
MSC (1991):
Primary 53C50
MathSciNet review:
1348150
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
A general theory of regularity for Lorentzian Busemann functions in future timelike geodesically complete spacetimes is presented. This treatment simplifies and extends the local regularity developed by Eschenburg, Galloway and Newman to prove the Lorentzian splitting theorem. Criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of Bartnik.
References:
- [AH]
- L. Andersson and R. Howard, Comparison and rigidity theorems in semi-Riemannian geometry, preprint.
- [B1]
- R. Bartnik, Regularity of variational maximal surfaces, Acta Mathematica 161 (1988), 145--181. MR 90b:58255
- [B2]
- ------, Remarks on cosmological spacetimes and constant mean curvature surfaces, Commun. Math. Phys. 117 (1988), 615--624. MR 89g:53094
- [BE1]
- J.K. Beem and Paul E. Ehrlich, Singularities, incompleteness and the Lorentzian distance function, Math. Proc. Camb. Phil. Soc. 85 (1979), 161--178. MR 80b:83040
- [BE2]
- ------, Global Lorentzian Geometry, Marcel Dekker, New York, 1981. MR 82i:53051
- [B+]
- J.K. Beem, P.E. Ehrlich, S. Markvorsen, and G.J. Galloway, Decomposition theorems for Lorentzian manifolds with nonpositive curvature, J. Diff. Geom. 22 (1985), 29--42. MR 87i:53093b
- [Bu+]
- R. Budic, J. Isenberg, L. Lindblom and P. B. Yasskin, On the determination of Cauchy surfaces from intrinsic properties, Comm. Math. Phys. 61 (1978), 87--95. MR 58:9089
- [E1]
- J.-H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), 295--323. MR 89f:53062
- [E2]
- ------, The splitting theorem for spacetimes with strong energy condition, J. Diff. Geom. 27 (1988), 477--491. MR 89f:53096
- [E3]
- ------, Maximum principles for hypersurfaces, Manuscripta Math. 64 (1989), 55--75. MR 90c:53134
- [EG]
- J.-H. Eschenburg and G.J. Galloway, Lines in spacetimes, Comm. Math. Phys. 148 (1992), 209--216. MR 93i:53064
- [EH]
- J.-H. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Glob. Analysis and Geometry 2 (1984), 141--151. MR 86h:53042
- [G1]
- G.J. Galloway, Splitting theorems for spatially closed space-times, Commun. Math. Phys. 96 (1984), 423--429. MR 86c:53042
- [G2]
- ------, Some results on Cauchy surface criteria in Lorentzian geometry, Illinois J. of Math. 29 (1985), 1--10. MR 86c:53043
- [G3]
- ------, Curvature, causality and completeness in spacetimes with causally complete spacelike slices, Math. Proc. Camb. Phil. Soc. 99 (1986), 367--375. MR 87h:53091
- [G4]
- ------, The Lorentzian splitting theorem without completeness assumption, J.Diff. Geom. 29 (1989), 373--387. MR 90d:53077
- [G5]
- ------, Some connections between global hyperbolicity and geodesic completeness, J. Geom. Phys. 6 (1989), 127--141. MR 90k:53102
- [G6]
- ------, The Lorentzian version of the Cheeger-Gromoll splitting theorem and its application to general relativity, Proc. Symp. Pure Math., vol. 54 (R. Greene and S.-T. Yau, eds.), Part 2, 1993, pp. 249--257. MR 94b:53102
- [GC]
- C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983), 523--553. MR 85h:53049
- [GR1]
- R. Geroch, Singularities in closed universes, Phys. Rev. Lett. 17 (1966), 445--447.
- [GR2]
- ------, Singularities, Relativity (M. Carmeli, S. Fickler and L. Witten, eds.), Plenum Press, New York, 1970, pp. 259--291. MR 51:2589
- [HE]
- S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973. MR 54:12154
- [N]
- R.P.A.C. Newman, A proof of the splitting conjecture of S.-T. Yau, J. Diff. Geom. 31 (1990), 163--184. MR 91h:53062
- [ON]
- B. O'Neill, Semi-Riemannian geometry, Academic Press, New York, 1983. MR 85f:53002
- [P]
- R. Penrose, Techniques of differential topology in relativity, Regional Conference Series in Applied Mathematics (no. 7), SIAM, Philadelphia, 1972. MR 57:8942
- [S]
- H.-J. Seifert, Global connectivity by timelike geodesics, Zs. f. Naturforsche 22a (1967), 1356--1360. MR 37:1149
- [Y]
- S.-T. Yau, Problem section, Ann. Math. Studies, vol. 102, ed. S.-T. Yau, 1982, pp. 669--706. MR 83e:53029
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (1991):
53C50
Retrieve articles in all Journals with
MSC (1991):
53C50
Additional Information:
Gregory
J.
Galloway
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email:
galloway@math.miami.edu
Arnaldo
Horta
Affiliation:
National Security Agency, Fort Meade, Maryland 20755-6000
Email:
ahorta@ix.netcom.com
DOI:
10.1090/S0002-9947-96-01587-5
PII:
S 0002-9947(96)01587-5
Received by editor(s):
September 9, 1994
Additional Notes:
The first author was partially supported by NSF grant DMS-9204372.
Copyright of article:
Copyright
1996,
American Mathematical Society
|
