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

TL;DR

This thesis studies the gradient flow of the length functional on embedded loops within scale manifolds, establishing compactness and convergence results akin to Morse theory, with implications for understanding geometric flows.

Contribution

It introduces a scale structure on the space of embedded loops, proving weak compactness and Floer-Gromov convergence for gradient flows related to curvature and length.

Findings

Proves weak compactness of gradient flow lines between geodesics.

Establishes Floer-Gromov convergence of certain gradient flows.

Visualizes gradient and curvature flows on various manifolds.

Abstract

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.