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Embeddedness, convexity, and rigidity of hypersurfaces in product spaces. (English) Zbl 1472.53031

The authors establish the following Hadamard-Stoker-type result: Let \(f: M^n\to H^n\times R\), \(n\ge 3\), be a complete connected hypersurface in a Hadamard manifold with positive definite second fundamental form, and let the height function of \(f\) have a critical point, then \(M\) is embedded and homeomorphic to \(S^n\) or \(\mathbb{R}^n\); furthermore, \(f(M)\) bounds a convex set in \(H^n\times \mathbb{R}\). Section 2 contains some notation and results which will be used afterwards. In Section 3, the authors prove Theorems 1–3 and Corollaries 1 and 2. Analogous results (Theorems 4 and 5) for hypersurfaces in warped product spaces \(\mathbb{R}\times_\rho H^n\) and \(\mathbb{R}\times_\rho S^n\) are proved in Section 4.

MSC:

53B25 Local submanifolds
53C24 Rigidity results
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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