Hyperbolicity of the time-like extremal surfaces in minkowski spaces

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)$$\partial \_tW+\sum \_{j=1}^{n}A\_j(W)\partial \_x\_jW=0,W:(t,x)\in {\mathbb{R}}^{1+n}\to W(t,x)\in {\mathbb{R}}^{n+m+\left(\genfrac{}{}{0ex}{}{m+n}{n}\right)},$$where each $A\_j(W)$ is just a $(n+m+(\genfrac{}{}{0ex}{}{m+n}{n}\left)\right)\times (n+m+(\genfrac{}{}{0ex}{}{m+n}{n}\left)\right)$ symmetric matrix dependinglinearly on $W$.