Light-likeness of boundaries of extremal surfaces of mixed type in physical space-times

In this paper, we prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space-time R1+n(n>1), in particular we improve Gu's theorem on the light-likeness of boundaries of extremal surfaces in R1+2 and prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space-times. As a consequence, we show that a curve moving in a physical space-time keeps its like-property and the boundary only exists when its world sheet at the initial time has light-like points. This implies that any extremal surface of mixed type is generated by an “initial curve of mixed type”.