Embeddedness, Convexity, and Rigidity of Hypersurfaces in Product Spaces

TL;DR

This paper proves new rigidity and convexity theorems for hypersurfaces in product spaces, characterizing their shape, embedding properties, and conditions under which they are spherical or cylindrical, extending classical results to more general ambient spaces.

Contribution

The paper introduces novel Hadamard--Stoker type theorems for hypersurfaces in product spaces, including conditions for embedding, convexity, and classification of hypersurfaces with constant curvature or mean curvature.

Findings

Hypersurfaces with positive definite second fundamental form and height function critical points are embedded and homeomorphic to spheres or Euclidean spaces.

Compact hypersurfaces with constant mean curvature are rotational spheres under certain conditions.

Hypersurfaces with positive semi-definite second fundamental form and no critical points in height are cylindrical and embedded.

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 $n\ge 3.$