zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Spacelike hypersurfaces with constant mean curvature in the steady state space. (English) Zbl 1162.53043

The paper studies complete space-like hypersurfaces with constant mean curvature H in the de Sitter space.

The first result is that if the surface lies between two concentric horospheres, then H=1. The second theorem assumes that the surface lies between a horosphere of center a and the light-like hyperplane orthogonal to a in the ambient Minkowski space (part of de Sitter space delimited by this hyperplane is the so-called steady state space). If, moreover the hypersurface has a future pointing mean curvature vector then a bound for H is given.

If the dimension of de Sitter space is 3, the conditions of both theorems imply that the surface is actually totally umbilical.

The first theorem is then extended to hypersurfaces embedded in particular Lorentzian manifolds. They are a generalization of those obtained by quotient of steady state space by isometries acting cocompactly on horospheres (“de Sitter cusps”).

There exist many results about space-like hypersurfaces with constant mean curvature in de Sitter space, see for example [S. Montiel, Constant mean curvature space-like hypersurfaces in de Sitter spaces, Suh, Young Jin (ed.) et al., Proceedings of the 9th international workshop on differential geometry, Taegu, Korea, November 12–13, 2004. Taegu: Kyungpook National University. 17-30 (2004; Zbl 1075.53071)]

MSC:
53C42Immersions (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics