Vol. 288, No. 2, 2017

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ISSN: 0030-8730
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 L3 , which can be viewed as a solution to the constant mean curvature equation over a convex domain Ω 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