Zero constant mean curvature in Minkowski space versus in Euclidean space

Question

There's a famous result in $\mathbb{R}^3$ which goes as: there are no compact minimal surfaces in $\mathbb{R}^3$.So the mean curvature cannot be zero in compact surfaces in $\mathbb{R}^3$. Now what's the intuition behind why that isn't true in the Minkowski space $E_1^3$ (also known as $\mathbb{L}^3$)?

Answer 1

The main problem is that there are no compact surfaces in $\Bbb L^3$ with constant causal type, at all. One way to see this is to consider the projections of points in the surface $M$ onto coordinate planes of distinct causal types; take points $p,q \in M$ realizing the maximum of those projections. So $T_pM$ and $T_qM$ have distinct causal types.