Biharmonic hypersurfaces in a conformally flat space

TL;DR

This paper investigates biharmonic hypersurfaces within conformally flat spaces, deriving their defining equations and identifying metrics that transform minimal hypersurfaces in Euclidean space into biharmonic ones in conformally flat spaces.

Contribution

It derives the biharmonic hypersurface equation in conformally flat spaces and characterizes metrics that convert Euclidean minimal hypersurfaces into biharmonic hypersurfaces.

Findings

Derived the biharmonic hypersurface equation in conformally flat spaces.

Identified metrics transforming minimal Euclidean hypersurfaces into biharmonic ones.

Included known biharmonic hypersurfaces as special cases.

Abstract

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}\delta_{ij}$ on the Euclidean space $\mathbb{R}^{m+1}$ so that a minimal hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})$ in a Euclidean space becomes a biharmonic hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in [Ou1] and [OT] as special cases.