In this paper, we show that an $L_1$-2-type surface in the three-dimensional hyperbolic space $H^3subset R^4_1$ either is an open piece of a standard Riemannian product $ H^1(-sqrt{1+r^2})times S^{1}(r)$, or it has non constant mean curvature, non constant Gaussian curvature, and non constant principal curvatures.

It is shown that the Orlicz–Lorentz spaces $\ell ^n_{M,a}$, $n\in \mathbb {N}$, with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if norm $\| \cdot \|_{M,a}$ satisfies certain Hardy-type inequalities. This includes embedding some Lorentz $\mathrm {d}^n(a,p)$. The approach based on combinatorial averaging techniques, a new result independent intere...