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  $n\ge 3.$