Study Plan: Curve Shortening Flow

For a while now I have wanted to dig into the details of Gage-Grayson-Hamilton work on the "curve shortening flow." I have several reasons for this: 

1. If I am to call myself a differential geometer, I will really have to learn some more about geometric analysis fundamentals. CSF seems like the most accessible version of a large class of geometric evolution equations which are very important. (For example, Perelman's work on the Poincaré conjecture is essentially a control over the singularities of the Ricci flow introduced by Hamilton.)
2. There is this big project I never finished that would likely go easier if I knew some more about these kinds of things. Maybe this would be a good way to get back into it.
3. Discretized versions of the problem seem to be a place for getting students involved in mathematics. I have tried lots of different angles for getting students involved in research, and I have learned the hard way that I need a class of problems that satisfy these requirements:
1. easy to get started (low floor)
2. possibility of exciting new developments (high ceiling)
3. I have to know something about it.
4. I am teaching differential geometry this term, and I like to take the opportunity when teaching a course to deepen my knowledge of the material in some significant way.
5. I am giving a talk at an undergraduate conference pretty soon, and I want to talk about these ideas. (I gave them an abstract that would push me to learn more.)
6. The stuff is just cool.

So, I have started reading the primary literature already. I will write about what I am learning in small pieces, so this is an opportunity for me to organize the ideas into a sequence of blog posts. 

The list below is limited a bit by my current state of knowledge. I know more details about the beginning of the list (items 1-7), and less about the end which will likely expand quite a bit. 

Here is what I think will happen:

1. Fundamentals on smooth plane curves: adaptations of the Frenet-Serret apparatus to 2d; some basic formulae for computation
2. convex plane curves, in particular the idea of a support function; new expressions for length and area
3. some isoperimetric inequalities (following a paper by Harley Flanders). Possibly several posts.
4. the idea of curve shortening flow
5. Gage's isoperimetric inequality
6. The evolution of the isoperimetric profile under CSF.
7. The Hausdorff metric on closed sets
8. CSF shrinking convex curves to "round points." Likely several posts. 
9. About parabolic PDEs and their solutions.
10. Grayson's whisker lemma and the case of embedded non-convex curves.
11. Curves on surfaces.

This project is mainly for my own benefit. I want to get this stuff into my head, and writing it really helps me. If you find it interesting, too, come along.

Evolutes

An Evolute of a Circular Helix

In a previous post, we studied the involute of a curve. Now we will reverse the direction of the problem: Given a regular curve $\alpha$, another curve $\beta$ is called an evolute of $\alpha$ when $\alpha$ is an involute of $\beta$.

This means that

1. $\beta(s) - \alpha(s)$ lies along the tangent to $\beta$ at $\beta(s)$.
2. $\beta'(s)$ is orthogonal to $\alpha'(s)$.

Our goal is to understand what we can say about evolutes.

Involutes

Let $\alpha, \beta : (a,b) \to \mathbb{R}^3$ be regular curves. We say that $\beta$ is an involute of $\alpha$ when

1. $\beta(t)$ lies on the tangent line to $\alpha$ at $\alpha(t)$, and 
2. the tangents to $\alpha$ and $\beta$ at $\alpha(t)$ and $\beta(t)$ are orthogonal.

In this post, we will explore the basics of what one can say about an involute of a given curve.

Generalized Helices: Curves of Constant Slope

A regular curve in space is said to have constant slope when its tangents make a constant angle with a given fixed line. Such a curve is also called a generalized helix, or even just a helix amongst the cognoscenti. The line is called the axis of the helix.


An analytic way of stating the condition is as follows: Let $A$ be a unit vector parallel to the axis $\ell$. Denote the parametrized curve by $X$ and its arclength parameter by $s$. We use standard notation for the Frenet-Serret apparatus. Then the definition means that
\[ \langle T, A \rangle = \cos \alpha . \tag{1}\]
Here $\alpha$ is the angle mentioned because $T$ and $A$ are both unit vectors.

Our goal is to describe the consequences of equation (1) for the shape of the curve.

Non-Unit Speed Curves: Curvature and Torsion

We have seen that any regular curve can be reparametrized by arclength. Of course, the process involves computing an integral and then describing the inverse of some increasing function. A little bit of Calculus tells us that it is theoretically possible, but it doesn't make it computationally feasible.

The first place we feel the pinch of this trouble is in computing curvature and torsion. We have convenient formulae for finding these functions for a unit speed curve, but if some guy from a dark alley steps out and opens his trenchcoat to reveal... lots of beautiful geometric space curves, and says, "Hey, buddy. Wanna buy a curve?", the chance that you get a curve which is already unit speed is pretty small. I mean, its probably not a real swiss timepiece, even if the guy claims its a Bernoulli Brachistochrone.

So, how can we deal with such a thing?

THE CHAIN RULE!

Now, we can get by for any single curve with the chain rule, but then we have to compute the whole Frenet-Serret apparatus, too. It would be much nicer to have compact and easy to compute formulae for the general case. That is our goal. (Note: I don't recall learning this when I first encountered the subject. I take the below from Struik [1].)

Parametrized curves: whence ``regularity''?

The first oddity one runs across in the study of the differential geometry of curves is the requirement that such a curve be regular. We shall describe the basic idea behind this little word and its significance.

Reparametrization and Arclength

Given all of the difficulties surrounding the parametrization of a curve, it would be good to have some geometrically meaningful way to do it. We want to have some way to choose a parametrization that actually has something to do with the shape of the curve. Fortunately, there is the parametrization by arclength. In this case, we use the distance along the curve from some fixed point as the parameter.

Differential Geometry of Curves: Why Parametrizations?

In differential geometry, we make a definite choice to study parametrized curves. The reasons for this are simple, but important. This choice also has some consequences for what kinds of things we will be able to handle. Here I want to discuss this basic idea.

Spherical Curves

When does a curve in space lie on a sphere? That is, is there a way to tell if some given curve actually lies on a sphere, without being given that information in advance?

My students asked this nice question, whose answer shows off the basic techniques of classical differential geometry very clearly. So, I figure it is helpful to share the answer, and also the way we found it in class.

The Tangent Spherical Image and another interpretation of Curvature

There is another slick little interpretation of the curvature. This one is both easier and harder to see: easier in that the argument for it is much shorter and doesn't use quite such an intricate argument; harder in that it lies so close to the surface it is just harder to notice.

As usual, we start with some $C^2$ regular curve $\gamma$ in $\mathbb{R}^3$, and we assume that it is parametrized by arclength with arclength parameter $s$. At this point, it can seem a trivial observation, but the important thing is that the tangent vector $T(s)$ has length which is always equal to $1$.

Now we step back and consider the assignment $\alpha: s \to T(s)$ as a function on its own terms, and let $\gamma$ fade into the background for a minute. For each input $s$, $T(s)$ is a vector in $\mathbb{R}^3$. Usually, one visualizes each of these vectors as attached to the corresponding points $\gamma(s)$, but we will just take them all based at the origin of $\mathbb{R}^3$. Because $||T(s)|| = 1$, the curve $\alpha$ lies on the unit sphere!

This curve $\alpha$ is called the (tangent) spherical image of $\gamma$.

An Interpretation of Curvature: Osculating Plane and Circle

To be very clear, this post follows Spivak Vol 2 pretty closely.
Suppose that we have some curve in space, and we have chosen a point $P$ on it. We are concerned with the following three questions:

1. There is no guarantee that the curve lies in a plane, but if there were a plane through $P$ which came closest to containing the curve, how would you find it?
2. Can we find a circle which serves as a good approximation to the curve near $P$?
3. Is there a simple and reasonable interpretation of the curvature of the curve at $P$?

Of course, the answers are all "yes." The plane is called the osculating plane, the circle is the osculating circle, and the curvature gets an interpretation from the size of the osculating circle. The goal of this post is to show that these objects exist and have real geometric significance.

Convenient Formulae for Curvature and Torsion

The Frenet-Serret apparatus is a useful and important way to deal with the geometry of differentiable curves, with the curvature and torsion helping us to measure the way a curve bends and twists around in space. Understanding $\kappa$ and $\tau$ is important and will be our key to the shape of the curve, so it will be important that we can compute them efficiently. The direct way is always available: simply compute the whole framing $\{T, N, B\}$ and find $\kappa$ and $\tau$ when computing some pair of $T'$, $N'$ and $B'$. But this is very time consuming. Can we find a faster way?

Problem: Find formulations for computing the curvature and torsion of a unit-speed curve $x(s)$ that involves only the derivatives $x'$, $x''$ and $x'''$ and easily computed linear algebra operations.

The Frenet-Serret Apparatus for Space Curves

The most important tool in classical differential geometry of curves is the orthonormal framing on the curve provided by the tangent, normal, and binormal vectors to the curve, and the Frenet-Serret equations which control the evolution of that framing.

Let's introduce what these things are, and how they help us define and compute the curvature and torsion of the curve. This will be very computational. We will fill in interpretations of the quantities we describe in other posts.