#### Vol. 288, No. 2, 2017

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A sharp height estimate for the spacelike constant mean curvature graph in the Lorentz–Minkowski space

### Jingyong Zhu

Vol. 288 (2017), No. 2, 489–509
##### Abstract

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 ${\mathbb{L}}^{3}$, which can be viewed as a solution to the constant mean curvature equation over a convex domain $\Omega \subset {ℝ}^{2}$. 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.

##### Keywords
height estimate, critical point, constant mean curvature, a priori estimates, Lorentz–Minkowski space.
##### Mathematical Subject Classification 2010
Primary: 35B38, 35B45, 35J93, 53C42, 53C50
##### Milestones
Received: 30 May 2016
Revised: 12 October 2016
Accepted: 14 December 2016
Published: 28 April 2017
##### Authors
 Jingyong Zhu School of Mathematical Sciences University of Science and Technology of China 96 Jinzhai Road, Heifei 230026 Anhui China