Global Existence for the Minimal Surface Equation on ${\mathbb{R}}^{1,1}$

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}{l}^{1,1}$ describing graphical time-like minimal surfaces embedded in ${\mathbb{R}}^{1,2}$ enjoy small data global existence for compactly supported initial data, using Christodoulou’s conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity.