Embeddedness, convexity, and rigidity of hypersurfaces in product spaces

Abstract

We establish the following Hadamard–Stoker-type theorem: Let \(f:M^n\rightarrow \mathscr{H} ^{\,\,\, n}\times \mathbb{R}\) be a complete connected hypersurface with positive definite second fundamental form, where \(\mathscr{H} ^{\,\,\, n}\) is a Hadamard manifold. If the height function of f has a critical point, then it is an embedding and M is homeomorphic to \(\mathbb{S}^n\) or \(\mathbb{R}^n.\) Furthermore, f(M) bounds a convex set in \(\mathscr{H} ^{\,\,\, n}\times \mathbb{R}.\) In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) as well. We apply these theorems to show that a compact connected hypersurface in \(\mathbb{Q}_{\epsilon}^{n}\times \mathbb{R}\) (\(\epsilon =\pm 1\)) is a rotational sphere, provided it has either constant mean curvature and positive-definite second fundamental form or constant sectional curvature greater than \((\epsilon +1)/2.\) We also prove that, for \(\bar{M}=\mathscr{H} ^{\,\,\, n} \,{\mathrm{or}} \,\, \mathbb{S}^n,\) any connected proper hypersurface \(f:M^n\rightarrow \bar{M}^n \times \mathbb{R}\) with positive semi-definite second fundamental form and height function with no critical points is embedded and isometric to \(\Sigma ^{n-1}\times \mathbb{R},\) where \(\Sigma ^{n-1}\subset \bar{M}^n\) is convex and homeomorphic to \(\mathbb{S}^{n-1}\) (for \(\bar{M}^n=\mathscr{H} ^{\,\,\, n}\) we assume further that f is cylindrically bounded). Analogous theorems for hypersurfaces in warped product spaces \(\mathbb{R}\times _\varrho \mathscr{H} ^{n}\) and \(\mathbb{R}\times _\varrho \mathbb{S}^{n}\) are obtained. In all of these results, the manifold \(M^{n}\) is assumed to have dimension \(n\ge 3.\)