Hyperpolic theory for the evolution of convex plane curves

This study consists of eight chapters.In the first chapter, we define the hyperbolic evolution problem of plane curves.In the second chapter, we state the fundamental theorem of the local theory of curves, different definitions of curvature of a curve, Frenet-Serre equations of the curves and further more we investigate the motion of a plane curve with respect to time variable and obtain its evolution equations.In the third chapter, we deal with the general evolution equation of embedded planar curves and obtain the evolution equations for the length of the curve and the area it bounds and prove that the tangential components of the evolution vector do not effect the length and area during the evolution process.Fourth chapter is devoted to proving that the evolving curve remain convex during the evolution process and the final shape is a circle in the Haussdorff metric if the initial curve is convex. In the fifth chapter, It is defined the normal evolution and shown that this property is preserved during the evolution. Moreover, in the end of chapter it is obtained a hyperbolic Monge-Ampere equation. In the sixth chapter, a hyperbolic normal curvature evolution is considered with an example and proved that after a finite time the evolution curve evolves into a single point. This chapter also includes some crucial results which support our assertion in the thesis. In the seventh chapter, we search the close relationship between the hyperbolic mean curve flow and the evolution equation for relativistic string in the Minkowski space time .Finally, the last chapter covers proof of the Theorem 1.1 as the result of the Example 6.1 and the theorems in the other chapters and a general evaluation of the thesis.