A SHARP HEIGHT ESTIMATE FOR THE SPACELIKE CONSTANT MEAN CURVATURE GRAPH IN THE LORENTZ-MINKOWSKI SPACE

Based on the local comparison principle of Chen and Huang (1982), 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 spacelike constant mean curvature graph in 3-dimensional Lorentz-Minkowski space L-3, which can be viewed as a solution to the constant mean curvature equation over a convex domain Omega subset of R-2. We get the uniqueness of critical points for such a solution, which is an analogue of a result of Sakaguchi (1988). 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.