# Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$

## TL;DR

This paper classifies biharmonic hypersurfaces in product spaces of Einstein spaces and real lines, revealing conditions under which they are minimal or vertical cylinders, and providing explicit classifications in specific cases.

## Contribution

It extends the understanding of biharmonic hypersurfaces by deriving new equations and classifications in product spaces involving Einstein spaces and real lines.

## Key findings

- Biharmonic hypersurfaces with constant mean curvature are either minimal or vertical cylinders.
- Derived explicit biharmonic equations in $S^m\times \mathbb{R}$ and $H^m\times \mathbb{R}$.
- Classified biharmonic hypersurfaces that are totally umbilical or semi-parallel for $m\ge 3$, including specific cases in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$.

## Abstract

In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of \cite{OW} and \cite{FOR}. We derived the biharmonic equation for hypersurfaces in $S^m\times \mathbb{R}$ and $H^m\times \mathbb{R}$ in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi-parallel for $m\ge 3$, and some classifications of biharmonic surfaces in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$ which are constant angle or belong to certain classes of rotation surfaces.