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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?

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    $\begingroup$ Wikipedia article has bunch of math/non-math applications:en.wikipedia.org/wiki/Curve-shortening_flow#Applications $\endgroup$ – Piyush Grover Apr 3 '17 at 21:55
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    $\begingroup$ Well, you cannot have it both ways. Either curves are so simple there is relatively little to study, or there are some deep and beautiful things to study about curves such as.... the curve shortening flow. $\endgroup$ – Lee Mosher Apr 4 '17 at 2:47
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    $\begingroup$ I'm surprised by the many up-votes. There was a time this sort of question was not OK. It's a kind of 'What's the deal with Topic X' type question. Have you "thoroughly searched for an answer before asking your question"? (quoted from the 'How to ask a good question' page). I think Lee Mosher's comment sort of nails it to be honest: I can paraphrase your first paragraph as "In studying Riemannian geometry, I am presented with lots of evidence that deep and interesting work is done on curve shortening flow". But then it's like you go on to ask "Can this really be true?" $\endgroup$ – Thompson Apr 4 '17 at 18:28
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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.

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  • $\begingroup$ Is it instrumental in the proofs of any particular theorems, or in the development of any particular theory? $\endgroup$ – Monstrous Moonshine Apr 3 '17 at 20:51
  • $\begingroup$ It was a way to try to gain understanding of the issues, which wind up being quite different in the curve setting. But again, the questions are beautiful in and of themselves. Why do you do mathematics? $\endgroup$ – Igor Rivin Apr 3 '17 at 20:55
  • $\begingroup$ I find the subject interesting myself, but I unfortunately now need to justify it to a non-mathematical audience... $\endgroup$ – Monstrous Moonshine Apr 3 '17 at 20:59
  • $\begingroup$ @MonstrousMoonshine Check out the edit. $\endgroup$ – Igor Rivin Apr 3 '17 at 21:12
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First, the curve-shortening process is quite beautiful in its own right:


          Spiral
  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.

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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).

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    $\begingroup$ This is a very nice argument, but, to be devil's advocate, it requires more regularity than many others, and perhaps more significantly, it is a triviality that an isoperimetric curve should be convex, so we can restrict to those, in which case the result is MUCH easier (Gage, not Grayson). $\endgroup$ – Igor Rivin Apr 4 '17 at 1:08
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    $\begingroup$ Gage observed the monotonicity of the isoperimetric ratio. $\endgroup$ – Paul Bryan Apr 4 '17 at 1:20
  • $\begingroup$ @IgorRivin By distance comparison arguments it is not hard to prove the main CSF result for arbitrary embedded curves. I think that proof is MUCH easier than Gage-Hamilton. $\endgroup$ – Paul Bryan Apr 4 '17 at 1:21
  • $\begingroup$ @IgorRivin You're absolutely correct, but it does go to show how natural curve-shortening flow is! I should also say that the convexity of an isoperimetric domain is only trivial on $\mathbb{R}^2$. It's non-trivial in higher dimensions or if you're not in Euclidean space. Perhaps this argument could suggest a higher-dimensional argument that is less trivial. Though, as far as I understand--correct me if I'm wrong here, mean curvature flow has not had much success with regards to isoperimetry. $\endgroup$ – Alec Payne Apr 4 '17 at 1:25
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    $\begingroup$ @AlecPayne See Felix Schulze's paper arxiv.org/abs/math/0606675 for partial results in higher dimensions. Taking the convex hull in higher dimensions may worsen the isoperimetric ratio, but taking the mean-convex hull improves it. The problem is as you say, that singularities may form for mean-convex mean curvature flow and he deals with this by level set methods but with a dimensions restriction of $n\leq 7$. $\endgroup$ – Paul Bryan Apr 4 '17 at 1:40
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You absolutely must consult the Wikipedia article on CSF.

Some interesting applications:

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