|
Geometric analysis of Lorentzian distance function on spacelike hypersurfaces
Author(s):
Luis
J.
Alías;
Ana
Hurtado;
Vicente
Palmer
Journal:
Trans. Amer. Math. Soc.
362
(2010),
5083-5106.
MSC (2010):
Primary 53C50, 53C42, 31C05
Posted:
May 25, 2010
MathSciNet review:
2657673
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain sharp estimates for the mean curvature of those hypersurfaces. Moreover, we also give a sufficient condition for its hyperbolicity.
References:
-
- 1.
- J. A. Aledo and L. J. Alías, On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz-Minkowski space, Manuscripta Math. 101 (2000), 401-413. MR 1751041 (2001a:53091)
- 2.
- J. A. Aledo and L. J. Alías, On the curvatures of complete spacelike hypersurfaces in de Sitter Space, Geometriae Dedicata 80 (2000), 51-58. MR 1762499 (2001f:53125)
- 3.
- J.K. Beem, P.E. Ehrlich and K.L. Easey, Global Lorentzian Geometry, Marcel Dekker, Inc., New York (1996). MR 1384756 (97f:53100)
- 4.
- J.K. Beem, P.E. Ehrlich, S. Markvorsen and G.J. Galloway, Decomposition theorems for Lorentzian manifolds with nonpositive curvature, J. Differential Geometry 22 (1985), 29-42. MR 826422 (87i:53093b)
- 5.
- G.P. Bessa and M.S. Costa, On cylindrically bounded
 -hypersurfaces of  , Differential Geometry and its Applications 26 (2008), 323-326. MR 2421710 - 6.
- Y.D. Burago and V.A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin (1988). MR 936419 (89b:52020)
- 7.
- Q. Chen, and Y.L. Xin, A generalized maximum principle and its applications in geometry, Amer. J. Math. 114 (1992), 355-366. MR 1156569 (93g:53054)
- 8.
- F. Erkekoglu, E. García-Río, and D. N. Kupeli, On level sets of Lorentzian distance function, General Relativity and Gravitation 35 (2003), 1597-1615. MR 2014496 (2005a:53119)
- 9.
- J.-H. Eschenburg, The splitting theorem for space-times with strong energy condition, J. Differential Geometry 27 (1988), 477-491. MR 940115 (89f:53096)
- 10.
- G.J. Galloway, Splitting theorems for spatially closed space-times, Comm. Math. Phys. 96 (1984), 423-429. MR 775039 (86c:53042)
- 11.
- G.J. Galloway, The Lorentzian splitting theorem without the completeness assumption, J. Differential Geometry 29 (1989), 373-387. MR 982181 (90d:53077)
- 12.
- R. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York (1979). MR 521983 (81a:53002)
- 13.
- A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135-249. MR 1659871 (99k:58195)
- 14.
- S. Markvorsen and V. Palmer, Transience and capacity of minimal submanifolds, GAFA, Geometric and Functional Analysis 13 (2003), 915-933. MR 2006562 (2005d:58064)
- 15.
- S. Markvorsen and V. Palmer, How to obtain transience from bounded radial mean curvature, Transactions of the Amer. Math. Soc. 357 (2005), 3459-3479. MR 2146633 (2006i:58050)
- 16.
- R.P.A.C. Newman, A proof of the splitting conjecture of S.-T. Yau, J. Differential Geom. 31 (1990), 163-184. MR 1030669 (91h:53062)
- 17.
- B. O'Neill, Semi-Riemannian Geometry; With Applications to Relativity, Academic Press, New York (1983). MR 719023 (85f:53002)
- 18.
- H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205-214. MR 0215259 (35:6101)
- 19.
- S. Pigola, M. Rigoli and A.G. Setti, Maximum principles on Riemannian manifolds and applications, Memoirs Amer. Math. Soc. 174, no. 822 (2005). MR 2116555 (2006b:53048)
- 20.
- B.Y. Wu, On the mean curvature of spacelike submanifolds in semi-Riemannian manifolds, Journal of Geometry and Physics 56 (2006), 1728-1735. MR 2240419 (2007e:53069)
- 21.
- S.T. Yau, Harmonic function on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201-228. MR 0431040 (55:4042)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2010):
53C50, 53C42, 31C05
Retrieve articles in all Journals with
MSC (2010):
53C50, 53C42, 31C05
Additional Information:
Luis
J.
Alías
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Email:
ljalias@um.es
Ana
Hurtado
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain
Email:
ahurtado@ugr.es
Vicente
Palmer
Affiliation:
Departament de Matemàtiques, Universitat Jaume I, E-12071 Castelló, Spain
Email:
palmer@mat.uji.es
DOI:
10.1090/S0002-9947-2010-04992-X
PII:
S 0002-9947(2010)04992-X
Keywords:
Lorentzian distance function,
Lorentzian index form,
spacelike hypersurface,
transience,
Brownian motion,
mean curvature,
function theory on manifolds.
Received by editor(s):
March 7, 2008
Posted:
May 25, 2010
Additional Notes:
This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010).
The research of the first author was partially supported by MEC project MTM2007-64504 and Fundación Séneca project 04540/GERM/06, Spain
The second author was supported by Spanish MEC-DGI grant No. MTM2007-62344, the Bancaixa-Caixa Castelló Foundation and Junta de Andalucía grants PO6-FQM-5088 and FQM325.
The third author was supported by Spanish MEC-DGI grant No. MTM2007-62344 and the Bancaixa-Caixa Castelló Foundation
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
