  This master thesis looks at the gradient flow of the length functional on
embedded loops. The space of embedded loops is endowed with a scale structure
so that the length functional becomes scale smooth. For certain underlying
manifolds, using the fact that the gradient flow of the length is the same as
the curvature flow, a weak compactness property for the gradient flow lines
between fixed critical points (geodesics) is proven. Also Floer-Gromov
convergence of certain gradient flow lines is achieved. These convergence
properties are very similar to those in Morse theory. Figures visualizing the
gradient/curvature flow on several manifolds are included.
