I've recently been studying Riemannian geometry with goal of studying and doing research in Ricci flow, however, I've been noticing that a lot of work in Riemannian geometry seems to be done in curvature shortening flows. Now, most people are familiar with the power of Ricci flow – besides being essential in proving the Poincaré Conjecture and the Sphere Theorem, Ricci flow is in general very useful for uniformizing metrics on 3-dimensional manifolds, and is additionally finding new uses in higher dimensions.
But, why do people care about curve shortening flow? Curves seem so simple that there would be relatively little to study (besides well known aspects such as the fundamental group, or holonomy). Why do people study curve shortening flow?
Best Answer
First, because it is a beautiful subject. Second, because it seemed at the time (early 1980s) like a good warm-up to higher dimensional curvature flows.
UPDATE For a non-mathematical audience, see the results and references on the Carpenter's Rule Problem.