Hyperbolicity of the time-like extremal surfaces in minkowski spaces

TL;DR

This paper proves that time-like extremal surfaces in Minkowski spaces can be described by a symmetric hyperbolic PDE system with a simple structure, specifically for graph cases.

Contribution

It establishes a symmetric hyperbolic PDE formulation for time-like extremal surfaces in Minkowski space, revealing a simple structure similar to the inviscid Burgers equation.

Findings

Describes the PDE system for extremal surfaces as symmetric hyperbolic

Shows the matrices depend linearly on the solution W

Provides a simple, Burgers-like structure for the equations

Abstract

In this paper, it is established, in the case of graphs, that time-like extremal surfaces of dimension $1+n$ in the Minkowski space of dimension $1+n+m$ can be described by a symmetric hyperbolic system of PDEs with the very simple structure (reminiscent of the inviscid Burgers equation)$$ \partial\_t W + \sum\_{j=1}^n A\_j(W)\partial\_{x\_j} W =0,\;\;\;W:\;(t,x)\in\mathbb{R}^{1+n}\rightarrow W(t,x)\in\mathbb{R}^{n+m+\binom{m+n}{n}},$$where each $A\_j(W)$ is just a $\big(n+m+\binom{m+n}{n}\big)\times\big(n+m+\binom{m+n}{n}\big)$ symmetric matrix dependinglinearly on $W$.