Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. (English) Zbl 0721.53055
The authors study the problem of prescribed mean curvature in Lorentz manifolds which have compact Cauchy surfaces and satisfy the timelike convergence condition. They prove that, given any smooth function \({\mathcal H}\) of \({\mathcal V}\) with causal past directed gradient and spacelike hypersurfaces \(M_+\), \(M_ -\) \((M_+\) lying strictly to the future of \(M_ -)\) there exists a spacelike hypersurface between \(M_ -\) and \(M_+\) with mean curvature \({\mathcal H}\), if \({\mathcal H}_{| M_ - /M_+}\) is estimated from above/below by the mean curvature of \(M_ - /M_+.\)
They also show that, given a Cauchy surface M and spacelike hypersurfaces \(M_+\), \(M_ -\) as above, for any y in M there exists a constant mean curvature surface passing through y, if supremum and infimum of the mean curvature of M are estimated by the mean curvatures of \(M_ -\) and \(M_+\). There also exists for any given reference surface \(\Sigma\) a constant mean curvature that encloses the same volume with \(\Sigma\) as M. The surfaces are constructed as asymptotic limits of a geometric evolution equation. Estimates on the rate of convergence are given.
They also show that, given a Cauchy surface M and spacelike hypersurfaces \(M_+\), \(M_ -\) as above, for any y in M there exists a constant mean curvature surface passing through y, if supremum and infimum of the mean curvature of M are estimated by the mean curvatures of \(M_ -\) and \(M_+\). There also exists for any given reference surface \(\Sigma\) a constant mean curvature that encloses the same volume with \(\Sigma\) as M. The surfaces are constructed as asymptotic limits of a geometric evolution equation. Estimates on the rate of convergence are given.
Reviewer: M.Kriele (Berlin)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |
Keywords:
prescribed mean curvature; Lorentz manifolds; compact Cauchy surfaces; timelike convergence condition; spacelike hypersurfaceReferences:
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