Lorentz Hypersurfaces satisfying $\triangle \vec {H}= \alpha \vec {H}$ with non diagonal shape operator

TL;DR

This paper investigates Lorentz hypersurfaces in pseudo-Euclidean space with complex eigenvalues and a specific differential equation, proving constant mean curvature under certain conditions on the shape operator.

Contribution

It establishes that Lorentz hypersurfaces with complex eigenvalues and up to five distinct principal curvatures have constant mean curvature.

Findings

Lorentz hypersurfaces with complex eigenvalues satisfy the differential equation with constant mean curvature.

The result applies to hypersurfaces with non-diagonal shape operators having at most five principal curvatures.

The study extends understanding of curvature properties in pseudo-Euclidean geometry.

Abstract

We study Lorentz hypersurfaces $M_{1}^{n}$ in $E_{1}^{n+1}$ satisfying $\triangle \vec {H}= \alpha \vec {H}$ with non diagonal shape operator, having complex eigenvalues. We prove that every such Lorentz hypersurface in $E_{1}^{n+1}$ having at most five distinct principal curvatures has constant mean curvature.