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

A Sharp Height Estimate for the Spacelike Constant Mean Curvature Graph in the Lorentz-Minkowski Space