zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Hypersurfaces of prescribed curvature in Lorentzian manifolds. (English) Zbl 1034.53064

For years the author, R. Bartnik, and J. Urbas have been considering the problem of finding a closed hypersurface of prescribed curvature F in a complete (n+1)-dimensional manifold N. That is, if Ω is a connected open subset of N, f is a positive function of class C 2,α , and F is a smooth, symmetric function defined in an open cone Γ n , then the problem is to find a hypersurface MΩ such that F M =f(x), where F M means that F is evaluated at the vector (κ i (x)) whose components are the principal curvatures of M. The author studied the case when N is a Riemannian manifold and F=H is the mean curvature. In [J. Differ. Geom. 43, 612–641 (1996; Zbl 0861.53058) and Math. Z. 224, 167–194 (1997; Zbl 0871.53045)] he proved the existence of closed strictly convex Weingarten hypersurfaces of a Riemannian manifold, provided there exist appropriate barrier hypersurfaces.

In this paper, the author obtains similar results for closed strictly convex spacelike hypersurfaces in a globally hyperbolic Lorentzian manifold. The class of curvature functions permitted in the Lorentzian case is somewhat smaller than in the Riemannian case. This is because in the Lorentzian case the Gauss equations give rise to a term in the equation for the second fundamental form with the opposite sign from that in the Riemannian case. The author proves his result by studying the corresponding curvature flow problem and proving convergence to a stationary solution with the aid of suitable a priori estimates.

53C42Immersions (differential geometry)
53C44Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
35J60Nonlinear elliptic equations
58J05Elliptic equations on manifolds, general theory