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
, which
can be viewed as a solution to the constant mean curvature equation over a convex
domain
.
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.
Keywords
height estimate, critical point, constant mean curvature, a
priori estimates, Lorentz–Minkowski space.
