Year: 2026
Type: preprint
Abstract: In this paper we investigate biconservative and biharmonic hypersurfaces in product spaces $L^m(\varepsilon)\times {\mathbb R}$, where $L^m(\varepsilon)$ denotes the space form ${\mathbb S}^m$ or ${\mathbb H}^m$ of constant sectional curvature $\varepsilon=1$ or $\varepsilon=-1$ respectively. We obtain their complete description in the case of isoparametric hypersurfaces and hypersurfaces of constant angle. Moreover, we prove some classification results for biconservative and biharmonic hypersurfaces for which the gradient of the mean curvature function is a principal direction. In this context we obtain new examples of complete, non-cylindrical, non-totally umbilical biconservative hypersurfaces in ${\mathbb H}^m\times {\mathbb R}$. Then we prove a rigidity theorem related to these examples. Finally, we prove a global non-existence theorem for complete proper biharmonic hypersurfaces in ${\mathbb H}^m \times {\mathbb R}$ such that the square of the mean curvature function is bounded and the Ricci curvature is bounded from below.
Source: arXiv (Cornell University)
Authors: Branding, V., Montaldo, S., Oniciuc, C., Ratto, A.
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Fondazione di Sardegna
OpenAlex ID: W7220997682