  In this paper, based on the local comparison principle in [12], we study the
local behavior of the difference of two spacelike graphs in a neighborhood of a
second contact point. Then we apply it to the constant mean curvature equation
in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness
of critical point for the solution of such equation over convex domain, which
is an analogue of the result in [28]. Last, by this uniqueness, we obtain a
minimum principle for a functional depending on the solution and its gradient.
This gives us a sharp gradient estimate for the solution, which leads to a
sharp height estimate.
