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

TL;DR

This paper provides a geometric proof of global existence for the minimal surface equation on , revealing its null structure and requiring fewer derivatives and milder decay assumptions than previous methods.

Contribution

It offers a new geometric proof that highlights the null structure of the equation, simplifying the argument and broadening the conditions for global existence.

Findings

Established global existence for the minimal surface equation on .

Revealed the null structure inherent in the equations.

Reduced the regularity and decay assumptions needed for the proof.

Abstract

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.