# Mean curvature flow and knot theory

I am wondering if the mean curvature flow of one-dimensional submanifolds of $$\mathbb{R}^3$$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-Milnor theorem.

For reference, the Fenchel theorem (1929) says that if $$c:S^1\to\mathbb{R}^3$$ is a smooth embedding, then the total curvature is at least $$2\pi$$. The Fary-Milnor theorem (1949/50) says that if $$c$$ forms a nontrivial knot, then the total curvature is at least $$4\pi$$.

Steven Altschuler ("Singularities of the curve shrinking flow for space curves", JDG 1991) showed that if $$c_t:S^1\to\mathbb{R}^3$$ is a one-parameter family of smooth embeddings satisfying the mean curvature flow, then $$\frac{d}{dt}\int_{S^1}|\kappa_t|\leq -\int_{S^1}\tau_t^2|\kappa_t|$$ where $$\tau_t$$ is the torsion of $$c_t$$. So if $$\int_{S^1}|\kappa_0|<4\pi$$ then certainly $$\int_{S^1}|\kappa_t|<4\pi$$ for all positive $$t$$. So it seems like one could hope for a proof of the Fary-Milnor theorem which is more or less directly analogous to Hamilton-Perelman's proof of the Poincaré conjecture or of the topological classification of closed 3-manifolds with nonnegative and positive scalar curvature.

The problem would rest on understanding the singularities of the mean curvature flow. Altschuler seems to have shown that a singularity of the mean curvature flow in this setting is characterized by blowup of the curvature (just like Ricci flow), and that a nontrivial tangent flow is given either by an Abresch-Langer solution or the grim reaper solution. This seems to be directly parallel to the Perelman-Brendle theorem which says that the analogous blowup limit of a finite-time singularity of a Ricci flow on a compact 3-manifold is either a quotient of shrinking round cylinders or the Bryant soliton, which was (in a weaker version) Perelman's breakthrough result.

So it seems like the key ingredients are there. Can they be put together? It seems like the basic problem is that I don't know what the "surgery" analogue might look like or how it might be relevant.

So, more generally, forgetting about Fary-Milnor theorem in particular, could one hope to use mean curvature flow for any sort of application in knot theory? Perhaps the proper analogue of the Hamilton-Perelman approach would decompose a given knot as a connected sum of prime knots, and would give some canonical representation thereof? This seems to be comparable to the geometrization conjecture, although the existence and uniqueness of such a decomposition seems to be already known in knot theory.

In order that I have a reasonably concrete question:

• Ricci flow with surgery on 3D compact manifolds is closely analogous to mean curvature flow of mean-convex surfaces in $$\mathbb{R}^3$$ (Brendle-Huisken). Is there an analogy, or a conjectural analogy, to mean curvature flow of curves in $$\mathbb{R}^3$$, or in 3-manifolds? Are there (conjectural) applications in knot theory?

I couldn't find any references on the web. I am aware of Perelman's use of mean curvature of curves in a Ricci flow background to show finite extinction time of Ricci flow on simply-connected 3-manifolds, but the only detailed version which I know to exist of this, Morgan-Tian's book, seems to have some basic errors (cf. Bahri "Five gaps in mathematics" and some followups on the arxiv)

• One obstacle for this is that in $\mathbb{R}^3$, curve shortening flow does not preserve the embedding of curves. Unlike in the plane, curves can cross along the flow which affects the knottedness. – Gabe K May 22 at 11:02
• It's also worth noting that the Abresch-Langer solutions have a saddle point property, in that small deformations of them generally do not flow to them but either several copies of a circle or develop a local kink (i.e. a grim reaper curve). Also, all the singularities are planar, so aren't knotted. – Gabe K May 22 at 11:39
• Thanks, I should have remembered your first point... so any good application would have to come with some precise understanding of the extrinsic distance function? The second point is certainly interesting, but is it so crucial? As far as I know the Hamilton-Perelman analysis didn't need to know anything about stability of singularities – Quarto Bendir May 22 at 21:39

This is really a comment that's too long to fit in 500 characters, but let me try to explain the obstacles I mentioned in the comments. Sorry for how large the pictures are. I'll edit this if I figure out how to resize them.

Curve shortening flow three dimensions can self- intersect, which is the main roadblock. However, there are a few other issues that seem challenging, even if we can overcome this first problem. I'll try to give some pictures (all of which are from Wikipedia) to demonstrate the challenge of understand the limits of spatial CSF from initial data.

Suppose we start with a knot and we somehow know a priori that it does not self intersect under CSF. The question is whether the limiting curve can be used as some sort of model for that knot. The issue we immediately run into is the result of Altschuler you mentioned; the limiting curves are planar, so cannot be knotted. To get around this, one might imagine that the solution is to consider the limiting curve along with some sort of crossing diagram. In some cases, this might work, but I suspect that most of the time it does not.

For instance, if we start with a trefoil knot, it might be the case that the limit is what happens on the left side of the following picture by Au [1], where the circle is covered twice. This is a trefoil with minimal total curvature, but it might not be what we hoped for. It's not the Abresch-Langer solution that genuinely looks like a flattened trefoil, but this is pretty much the best case scenario otherwise. Even here, I don't think this is what happens generically.

For a less ideal example, suppose we start with a figure-8 knot with rotational symmetry, as shown below.

This also curvature at least curvature at least $$2 \pi$$. However, here the "minimal diagram" looks instead like this (rotated by 90 degrees).

This is not going to be the limit of CSF, which tells us that for this initial condition, either CSF has a local singularity or else becomes unknotted. It seems to be the latter, but a proof would take some careful analysis.

Somewhat related to all of this is that we can't easily rule out Type 2 singularities, which are essentially local kinks. In two dimensions, surgery might be useful for local singularities because they only happen when the curve crosses itself. If you cut around an intersection to make two curves, you expect to be able to continue the flow. For spatial curves, it's much less clear when Type 2 singularities emerge. Perhaps there is a version of Grayson's theorem that gives sufficient conditions for an initially embedded curve to develop a Type 1 singularity, but I'm not aware of it. There is a version if the curve is contained on the surface of a sphere, but that's not directly relevant here.

Au, Thomas Kwok-Keung, On the saddle point property of Abresch-Langer curves under the curve shortening flow, Commun. Anal. Geom. 18, No. 1, 1-21 (2010). ZBL1217.53067.