Some Special Extremal Surfaces In Reissner-nordstrom Spacetimes

This paper concerns some special extremal surfaces in the Reissner-Nordstr¨om space-time.Firstly, we introduces extremal surfaces and some properties of the Reissner-Nordstr¨om spacetime, and then give the main results and organization of this thesis. Secondly, we study the light-like extremal surfaces in Reissner-Nordstr¨om spacetimes. It is interesting to ?nd that under a di?eomorphic transformation of variables, the light-like extremal surfaces can be described by a system of nonlinear geodesic equations. Particularly, we investigate the light-like extremal surfaces in Reissner-Nordstr¨om spacetime in detail and obtain two special solutions. Finally, we investigate the motion of relativistic strings in the Reissner-Nordstr¨om space-time. As a general framework,we ?rst analyze the basic equations for the motion of a p-dimensional extended object in a general enveloping space-time(N, ?g), which is a given Lorentzian manifold, and then particularly investigate the interesting properties enjoyed by the equations for the motion of relativistic strings in the Reissner-Nordstr¨om space-time. Based on this, under suitable assumptions, we prove the global existence of smooth solutions of the Cauchy problem for the equations for the motion of relativistic strings with small arc length in the Reissner-Nordstr¨om space-time.