The Equations For Time-like Extremal Surfaces In The Minkowski Space R～(1+n)

The study of extremal surfaces is of substantial interest in mathematics, general relativity and string theory. It plays an important role in particle physics, fluid mechanics, electromagnetism as well as in the theory of black hole. In particular, the time-like extremal surfaces can aptly illustrate the motion of relativistic strings in the Minkowski space R¹⁺ⁿ, rendering it necessary for us to investigate the inherent nature and behavior of these surfaces. In this paper, we first introduce the background of the extremal surfaces, including both the Euclidian case and the Minkowski case. Based on this, we present its developmental results in the chronological order, where a lot of geometrical properties have been obtained and a variety of physical phenomena have been explained. Then by geometrical and variational methods we derive two versions of the equations for time-like extremal surfaces in the Minkowski space R¹⁺ⁿ, and prove their equivalence. In particular, we illustrate the relation between our equations and Kong's equation for the time-like extremal surfaces in the Minkowski space R¹⁺ⁿ.