# Geodesic Loopiness: An Entirely Unnecessary Post

As I understand it, Mean Curvature Flow / Curve Shortening Flow was originally introduced as a means of producing and analyzing closed geodesics.  Significant progress towards this goal was made throughout the decade 1980-1990 and owes much to the work of Gage, Grayson, Hamilton, Gluck, and Uhlenbeck (among others).  Even better, their work inspired a number of new ideas and questions, (arguably laying the groundwork for the flows Perelman adjusted and used to solve the Poincare conjecture).  To borrow a line from one of Dr. Gage’s talks, “the 80s were a good time for curvature flows.”

One such inspired question is this: Which curves exhibit self-similarity under this flow?  The first person to tackle this problem appears to be the celebrated physicist William W. Mullins.  He used mean curvature flow to model the motion of idealized grain boundaries in a 1956 paper, producing families of curves that either shrink of expand as they evolve.

Of course, the simplest example of a curve that shrinks under mean curvature flow is a circle.  In 1986, Abresch and Langer classified all curves with this shrinking property, curves heretofore referred to as “shrinkers.”  The vast majority of these shrinkers are stinkers, too, as they tend to be annulus-filling messes.  Only a countable sub-collection behave nicely, like the ones pictured below.

Closed Shrinkers

The astute reader will note that these examples are rosette-like curves that live in various annuli, and the insanely astute reader might guess that the ratio $p/q$ that compares the number of petals to the curve’s turning number depends transcendentally on the maximum radial value and can only take values in the interval $\left[\frac{1}{2},\frac{1}{\sqrt{2}}\right].$  Indeed, such a reader would be correct.

Abresch and Langer’s methods involve differential/Riemannian geometry, as well as a few ad hoc techniques and computations relegated to appendices.  A conjecture from their work stipulates that these shrinkers are certain kinds of saddle points, ones that lie “in between” circles and curves with cusps.  This was affirmatively resolved in 2001 by Thomas Kwok-Keung Au, and curves featuring simultaneous types of similarity (shrinkers and expanders that also rotate) have recently been constructed by Halldorsson.

Most of the work in this area relies if not heavily on PDE theory then at least non-trivially, including the work of Au and Halldorsson.  Interestingly enough, though, much of this dependency (and use of other “high level” mathematics) can be removed from Abresch and Langer’s original classification, so much of it, in fact, that their results become accessible to undergraduate math majors (relying upon elementary definitions from differential geometry or vector calculus, basic results in ODE theory, and calculus).

Au’s work contains the key to this simplification, a nifty geometric function called the support function.  Given a hypersurface $X \subset \mathbb{R}^{n+1}$ the support function of $X$ is given by $h = \langle X, N \rangle$ where $N$ is a unit normal.  For planar curves the support function is related to (signed) curvature via the equation

$k = -\frac{1}{h+h''}.$

Of course, mild restrictions apply.  Most notably, the curve $X(u)$ must be locally convex so that it can be regularly parameterized by its unit normal $N(u) = e^{iu}$.  Assuming this and after applying a suitable rescaling, one discovers that $X(u)$ is a shrinker if and only if

$h'' = \frac{1}{h} - h.$

In fact, as noted in Au’s paper, for any locally convex curve evolving by Mean Curvature Flow, $h$ must satisfy the more general partial differential equation

$h_t = -\frac{1}{h+h_{u\,u}}$

where $h = h(u,t)$ with $u$ parameterizing the unit normal and $t$ representing time.

St. Mary’s math major Joshua Kaminsky and I spent the better part of the 2011/2012 academic year solving and studying the above ordinary differential equation (which can be obtained by seeking separable solutions to the above PDE).  We were able to derive the Abresch-Langer classification, but we stumbled across quite a few new ideas and questions, too.

Before continuing, though, I should point out that I did not arrive at this problem honestly.  That is to say that I was not thinking about mean curvature flow when this problem occurred to me.  Rather, I came across it after attending various talks by Frank Morgan and his students.  The equation $k+h = 0$ also shows up when seeking “weighted geodesics”  (more specifically, but admittedly less clear, “Gaussian-weighted geodesics”).  Indeed, the quantity $k+h$ is known as “weighted curvature.”

So-called Gaussian-weighted geodesics are curves that (locally) minimize

$\int \! e^{-r^2/2} \, ds$

This expression is nothing more than Euclidean arc-length that has been weighted by the Guassian (like) density $e^{-r^2/2}$ (it behooves me to mention that the $1/2$ factor is inconsequential; it corresponds to my choice of rescaling).

There is much more to share on this topic, and I intend to do so in upcoming posts, but I’m nearing a good stopping point now.  A few notes before concluding, though.

First, Dr. Morgan’s REU students have made much progress on studying these (and other) “weighted” creatures, an area of study called “Manifolds with Density” (check out this paper and this one for starters).  This casts our previous examples in a new light, making the unit circle and the more complicated but beautiful Abresch-Langer curves instances of Gaussian-weighted geodesics.  In fact, Abresch and Langer’s original work proved an equivalent version of Dr. Morgan’s students’ conjecture (that was later proved) claiming that the circle is the only embedded Gaussian-weighted geodesic.

Second, Josh and I have a conjecture that constant weighted-curvature curves evolve through other constant weighted-curvature curves under the standard mean curvature flow, but they do so towards weighted geodesics.  This aligns nicely with the previously mentioned conjecture made by Abresch and Langer and later verified by Au.  We have also come across some interesting and strange findings in our attempts to classify all constant weighted-curvature curves.  For instance, the bounds on that geometric ratio $p/q$ do not appear to hold.

Third, a 2012 St. Mary’s Undergraduate Research Fellow by the name of David Rice is currently wrapping up his exploration of some planar curves that exhibit translational self-symmetry under weighted curvature flow.  These are curves that evolve according to the equation

$X_t = (k+h)N.$

After that is said and done we plan to tinker around with some of the constant weighted-curvature curve problems alluded to above.

Fourth, I have not even touched upon the current and massive amount of work being done on shrinking surfaces.  While much PDE theory informs these results, geometric-oriented progress is currently under way (perhaps most notably by Colding and Minicozzi).

So, again I repeat, there’s a lot more to say on this!  I hope to sketch out some explanations and ideas in future posts, all culminating in a few submitted papers, perhaps even a talk or two.  For now, though, I’d like to reflect on a strange or unexpected loop that (hopefully) ties much of this together.

Here goes.

The hunt for geodesics involves a journey through other (specifically weighted) geodesics, which sounds a bit too self-referential for my liking … at least at first glance.  Its perfectly harmless, of course, and quite ingenious when taken all together.

1. Evolving a closed curve by mean curvature flow will decrease its length (hence the alternate name “curve shortening flow”), and so if the curve isn’t destroyed in the process, it becomes a closed geodesic.
2. The question then shifts to one every destined-to-die victim in a horror movie asks, “What could go wrong?”  That is, exactly how can a curve be destroyed?  Curves flowing in a surface develop singularities pretty much how planar curves do, and so the question can be answered by analyzing the destruction of planar curves.  It can be shown that when a closed planar curve dies it does so very much how a circle dies.  That is, in the final moments of the curve’s life, it looks very much like a shrinker.  Hence, if we can understand shrinkers, we can (better) understand closed geodesics.
3. Of course, shrinkers are themselves geodesics (with respect to a conformally Euclidean metric whose scaling factor involves a Gaussian density), and so the whole process can be repeated again.  One attempts to find weighted geodesics by looking for shrinkers under weighted mean curvature flow (the same one David Rice is studying).

Very loopy indeed.

Except that its not.  Not in the way you might think, at least.  You’d expect this process to continue ad infinitum, but it turns out that a shrinker with respect to weighted-curvature flow was actually a shrinker (or maybe an expander) with respect to standard curvature flow all along (albeit with a change of scaling factor).

Not convinced?  Here is the simple computation.  Assume $X(u)$ is a shrinker with respect to weighted curvature flow, i.e.

$c'(t)X(u) = \frac{\partial X(u,t)}{\partial t} = (k(u,t)+h(u,t))N(u)$

where $c(t)$ is a smooth function.  Evaluating this expression at $t=0$, dotting with $N(u)$, and letting $c'(0) = C$ produces the equation

$Ch(u) = \langle CX(u), N(u)\rangle = k(u)+h(u)$

which can be simplified to $k + (1-c)h = 0$.  When $1-c > 0$ this corresponds to old shrinkers, and when $1-c < 0$ we have an old expander.  In either event, the game is over.  We’ve “looped” back to our starting point.

So, to summarize all this loopiness about geodesic loops yet one more time: shrinkers help us understand how curves die (under mean curvature flow), in turn shedding light on geodesics.  Similarly, then, “meta-shrinkers” should help us understand how curves die (under weighted mean curvature flow), shedding light on weighted geodesics.  But meta-shrinkers are old shrinkers (or expanders).  Hence, they often explain themselves!

All of this goes to show you how entirely unnecessary this post was.  After all, its about something completely self explanatory.