Why do people study curve shortening flows? 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?
 A: 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
A: First, the curve-shortening process is quite beautiful in its own right:

          


 
YouTube video showing self-intersection is avoided.

Second, curve-shortening is a method for proving the existence of simple closed geodesics on
a topological $2$-sphere. See The theorem of the three geodesics.
A: 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.
A: As Joseph O'Rourke said, curve-shortening flow can be used to prove the theorem of the three geodesics. If a simple closed curve on a Riemannian surface evolves by curve-shortening flow and does not collapse to a point in finite time, then it smoothly approaches a geodesic. This is a result of Grayson.
Curve-shortening flow can also be used in a really beautiful way to prove the isoperimetric inequality in two dimensions for closed embedded curves. This works more generally on Riemannian surfaces, but it's really neat and tidy for $\mathbb{R}^2$. Edit: Gage originally showed that the quantity $\frac{L^2}{4\pi A}$ is non-increasing with respect to this flow.
Consider a closed embedded initial curve $\gamma_0$ (which we give a positive orientation), and let $\gamma_t$ be the solution to curve-shortening flow. By the Gage-Hamilton-Grayson theorem, $\gamma_t$ remains embedded, is smooth, and shrinks to a round point in finite time, i.e. the curve becomes rounder as it shrinks to a point and "converges" to a circle in a suitable sense. This is truly a remarkable theorem if you look at the spiral posted by Joseph O'Rourke--it's not obvious that such a thing should unravel and become convex. It's also remarkable considering how finnicky and prone to singularities mean curvature flow and Ricci flow are (in higher dimensions at least). 
Now, we have the following evolution equations for curve-shortening flow for the length $L$, area $A$, and curvature $k$ of the curve: 
$$\frac{\partial L}{\partial t} = - \int_{\gamma_t} k^2\, ds$$
$$\frac{\partial A}{\partial t} = -\int_{\gamma_t} k\, ds = -2\pi$$
The integrals are taken with respect to arc length, and the second equality in the second line follows from the fact that we are considering a positively-oriented simple closed curve so its winding number is 1. The first equation shows that the length is non-increasing, and in fact curve-shortening flow can be thought of as the flow of curves that decreases length "fastest" or "most efficiently". Using the above evolution equations, we can compute
$$\frac{\partial}{\partial t}(L^2 - 4\pi A) = -2L \int_{\gamma_t} k^2\, ds + 4\pi \int_{\gamma_t} k\, ds$$
$$= -2L \int_{\gamma_t} k^2\, ds + 2(2\pi) \int_{\gamma_t} k\, ds$$
$$=-2L \int_{\gamma_t} k^2\, ds + 2 \big(\int_{\gamma_t} k\, ds\big)^2$$
$$\leq -2L \int_{\gamma_t} k^2\, ds +2\big(\int_{\gamma_t} \, ds\big)\big(\int_{\gamma_t} k^2\, ds \big) $$
$$= -2L \int_{\gamma_t} k^2\, ds + 2L \int_{\gamma_t} k^2\, ds = 0$$
Thus, $L^2 - 4\pi A$ is non-increasing in time with respect to curve-shortening flow. As we mentioned before, closed embedded curves shrink to round points, so $L^2 \to 4\pi A$, since $L^2 = 4\pi A$ for circles. So, $L^2 - 4\pi A \to 0$. However, since $L^2 - 4\pi A$ is non-increasing in time, this means that for $\gamma_0$, $L^2 > 4\pi A$. This is the isoperimetric inequality in two dimensions for closed embedded curves.
Also, note that we may conclude that $L^2 = 4\pi A$ if and only if the curve is a circle. In the above calculation, we only applied Cauchy-Schwarz which is sharp exactly when $k$ is constant. For a closed embedded curve, this can only be a circle, by the fact that the curvature determines a curve up to isometry (fundamental theorem of planar curves). 
