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A hypersurface $ M^n $ in the Lorentz-Minkowski space $\mathbb{L}^{n+1} $ is called $ L_k $-biharmonic if the position vector $ \psi $ satisfies the condition $ L_k^2\psi =0$, where $ L_k$ is the linearized operator of the $(k+1)$-th mean curvature of $ M $ for a fixed $k=0,1,\ldots,n-1$. This definition is a natural generalization of the concept of a biharmonic hypersurface. We prove that any $ L_k $-biharmonic surface in $ \mathbb{L}^3 $ is $k$-maximal. We also prove that any $ L_k $-biharmonic hypersurface in $ \mathbb{L}^4 $ with constant $ k$-th mean curvature is $ k $-maximal. These results give a partial answer to the Chen's conjecture for $L_k$-operator that $L_k$-biharmonicity implies $L_k$-maximality.

In this study, we present two new regularity criteria based on pressure and its gradient to the Cauchy problem of the 3D magneto-micropolar system in anisotropic Lorentz spaces.

Dirac cone can be tilted in condensed matter setting. As a result of tilt, the Lorentz symmetry is reduced to what we call tilted-Lorentz symmetry. In this paper we derive the tilted-Lorentz transformations that leave a world with tilted Dirac cone invariant.

In this manuscript, we study the Lorentz hypersurfaces of the Lorentz 5-pseudosphere (i.e. the pseudo-Euclidean 5-sphere) $S^5_1$ having three distinct principal curvatures. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every submanifold is minimal. We consider an advanced version of the conjecture on Lorentz hypersurfaces of $S^5_1$. We present an affirmative answer to the extended conjecture on Lorentz hypersurfaces with three distinct principal curvatures.