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This master thesis looks at the gradient flow of the length functional onembedded loops. The space of embedded loops is endowed with a scale structureso that the length functional becomes scale smooth. For certain underlyingmanifolds, using the fact that the gradient flow of the length is the same asthe curvature flow, a weak compactness property for the gradient flow linesbetween fixed critical points (geodesics) is proven. Also Floer-Gromovconvergence of certain gradient flow lines is achieved. These convergenceproperties are very similar to those in Morse theory. Figures visualizing thegradient/curvature flow on several manifolds are included.

The curve shrinking flow, compactness and its relation to scale manifolds