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 $\bar M=\mathscr H^n$ 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 $\mathbb R\times_\rho\mathscr H^n$ and $\mathbb R\times_\rho\mathbb S^n$ are obtained. In all of these results, the manifold $M^n$ is assumed to have dimension $n\ge 3.$