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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 $\left(n+1\right)$-dimensional manifold $N$. That is, if ${\Omega }$ is a connected open subset of $N$, $f$ is a positive function of class ${C}^{2,\alpha }$, and $F$ is a smooth, symmetric function defined in an open cone ${\Gamma }\in {ℝ}^{n}$, then the problem is to find a hypersurface $M\in {\Omega }$ such that $F\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mid }_{M}=f\left(x\right)$, where $F\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mid }_{M}$ means that $F$ is evaluated at the vector $\left({\kappa }_{i}\left(x\right)\right)$ 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.

MSC:
 53C42 Immersions (differential geometry) 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35J60 Nonlinear elliptic equations 58J05 Elliptic equations on manifolds, general theory