TITLE: Why do people study curve shortening flows?
QUESTION [19 upvotes]: 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?

REPLY [8 votes]: You absolutely must consult the Wikipedia article on CSF.
Some interesting applications:

Isoperimetric inequality in the place (noted in the answers here already)
Closed geodesic theorems (noted in the answers here already)
Determination of isoperimetric regions in paraboloids of revolution: A new isoperimetric comparison theorem for surfaces of variable curvature
Used in the proof of the Poincaré conjecture! Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
Evolution of interfaces in variety of applications: Applications of CSF
In particular, CSF of networks models two-dimensional, multi-phase systems (original Mullins problem): Evolution of networks with multiple junctions