Biconservative Lorentz Hypersurfaces with at Least Three Principal Curvatures

Biconservative submanifolds, with important role in mathematical physics and differential geometry, arise as the conservative stress-energy tensor associated to the variational problem of biharmonic submanifolds. Many examples of biconservative hypersurfaces have constant mean curvature. A famous conjecture of Bang-Yen Chen on Euclidean spaces says that everybiharmonic submanifold has null mean curvature. Inspired by Chen conjecture, we study biconservative Lorentz submanifolds of the Minkowski spaces. Although the conjecture has not been generally confirmed, it has been proven in many cases, and this has led to its spread to various types of submenifolds. As an extension, we consider a advanced version of the conjecture (namely, $L_1$-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean space $\mathbb{M}^5 :=\mathbb{E}^5_1$ (i.e. the Minkowski 5-space). We show every $L_1$-biconservative Lorentz hypersurface of $\mathbb{M}^5$ with constant mean curvature and at least three principal curvatures has constant second mean curvature.