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The half-space property and entire positive minimal graphs in \(M \times \mathbb{R}\). (English) Zbl 1291.53075

Recently, many authors have studied surfaces and hypersurfaces of a product of a space form with constant sectional curvature and the real line. In the present paper the authors extend this study to hypersurfaces in the product of a complete \(n\)-dimensional manifold and the real line.
In particular they investigate conditions under which a properly immersed hypersurface is necessarily a slice, i.e. the product of the complete \(n\)-dimensional manifold with a point. If this is true they say that \(M\) has the half space property. They prove that this is true if 0.6 cm
1.)
\(M\) is a complete recurrent Riemannian manifold with bounded sectional curvatures, or
2.)
\(M\) is a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded from below and with positive height function.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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