# Biconservative Lorentz hypersurfaces in $\mathbb{E}_{1}^{\lowercase{n}+1}$ with complex eigenvalues

## TL;DR

This paper proves that biconservative Lorentz hypersurfaces with complex eigenvalues in Lorentzian space have constant mean curvature, and biharmonic ones are minimal, confirming Chen's conjecture in this setting.

## Contribution

It classifies biconservative and biharmonic Lorentz hypersurfaces with complex eigenvalues, establishing their geometric properties and confirming Chen's conjecture.

## Key findings

- Biconservative Lorentz hypersurfaces with complex eigenvalues have constant mean curvature.
- Biharmonic Lorentz hypersurfaces with complex eigenvalues are minimal.
- The results affirm Chen's conjecture on biharmonic submanifolds.

## Abstract

Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.