Math 110.745: Introduction to Curvature Flows, Fall 2020

Course description:
Geometric flows are important tools in geometry, topology and physics and have become an intrinsically interesting topic in their own right. Spectacular achievements include their application to the 3D Poincare conjecture and Thurston's geometrization conjecture and the Penrose conjecture in mathematical general relativity. In this course we will focus on the mean curvature flow, which has seen intense activity in recent years and is well represented in the department -- in particular by Professors Bernstein and Spruck.

Who, where, when:
Instructor: Alex Mramor, amramor1 'at'
Office hours: Zoom, by appointment

Prerecorded lectures will be posted to blackboard to be watched on your own time. I'll probably make 2-3 lectures for each week (I did this for the first month, but everyone seems busy/distracted (maybe for COVID related reasons) so I decided to slow down) a lecture or two every couple of weeks.

We will use many sources in this course; they (and maybe sometimes additional reading) will be listed below in the schedule.

Grading: Although the mean curvature flow is a great flow to study, there are other interesting flows as well. Your grade in this course will be based off a reading project (to be started in the early stages of the course) culminating in an essay or miniseries of lectures on a geometric flow/application of your choosing (after discussing with me).

Course schedule (updated often!):

Week 1:
This week we'll start off with an introductory lecture to flows and a couple sketches showing how they can be used to actually prove something. The next lecture is a crash course on the submanifold geometry, along with a proof of the short time existence of the flow by DeTurck's trick.

Reading and sources: There is a very recent book all about extrinsic curvature flows (of which the mean curvature flow is one) by Andrews, Chow, Guenther, and Langford which is sure to be an important resource to the field in the following years. The appendix has some nice material on ``standard parabolic theory'' which we glide over in lecture when discussing short time uniqueness.

When I was first learning about the mean curvature flow, Mantegazza's book was very helpful, and I'll surely draw some material from it later in the course.

If you want to quickly brush up on submanifold geometry and Riemannian geometry, John Lee's book introduction to curvature is a short and plesant read.

There are also many notes from classes by leaders in the field: I've seen nice class notes for the courses of Haslhofer, Hershkovits, Schulze, and White for example. For DeTurcks trick, I first read about it in Christos Mantoulidis' undergrad(!) thesis that you can find on his website -- its very nicely written and worth taking a look at.

Week 2:
This week we calculate the evolution of various geometric qunaities and discuss the ODE comparison principle. In the first lecture I essentially repeat the calculation of the first variation of area and explain why the mean curvature flow is the gradient of the area functional. Apropos of that I say a few words about minimal surfaces and using the MCF to find them (btw, a project on the 3 geodesics theorem or the Thomas-Yau conjecture would be good). There are many interactions between mean curvature flow and minimal surface theory, and ideas and techniques from it have had an impact on the development of the field. Here is a link to the now probably standard book ``A course in minimal surfaces'' by Colding and Minicozzi, which if you're in a book buying mood I'd highly recommend.

These minimal submanifold notes by Cheng, Li, and Mantoulidis from a class Rick Schoen taught were also really eye opening for me when I first encountered them. The choice of topics is superb; relating to our class positive isotropic curvature is like 2-convexity for the Ricci Flow, and as such Brendle has done a lot of recent work on the Ricci flow with surgery -- giving an outline might be a nice project. There is also a Ricci flow proof of the positive mass theorem in dimension 3 which might be good to report on as well (both subjects being mentioned in the notes).

In lecture 2 I show Simons identity -- I spent a lot more time on it than Mantegazza does but I think its worth dwelling on just a little bit and I suggest you go through the calculation on your own (either before or after watching the video). Something I forgot to say at the end: after all the calculating in the frame adapted at p you can go through pretty easily and ``reinsert'' the metric g if you keep track of where writing g_{ij} = \delta_{ij} was used and is a good quick exercise. To justify calling it a ``Bochner type formula,'' note the classical Bochner formula is a formula relating the Laplacian of a gradient term with the gradient of a lapacian term (and swtiching these of course gives curvature terms). Thinking of H as Laplacian of the position vector shows how they are alike because they answer similar questions (in their particular context): what types of curvature terms exactly does one pick up when commuting covariant derivatives of a certain quantity?

Minimal varieties in Riemannian manifolds is the landmark paper where Simons identity first appeared: in this paper Simons showed (amongst other things) that graphical minimal surfaces in R^{n+1} are planar up to n = 7. A bit later on it was found to be false in higher dimensions though by Bombieri, Giusti, and DeGiorgi. What's special about n = 8? As far as I know its still a bit myseterious.

Finally in the third lecture I calculate out the evolution equations for some curvature quantities, and then explain the ODE comparison/maximum principle. I wrap up with a couple first consequences (we'll see more ''first consequences'' soon though -- you can get a lot of milage out of the principle) of it: a doubling time estimate for the curvature and preservation of mean convexity.

Week 3:
This week there are just two lectures since I had something come up this week. The first lecture we compute (roughly -- the reaction terms we just leave in ''rough'' form since thats all we need of them) the evolution of the gradients of the curvature, the upshot being that if the second fundamental form is bounded on some time interval, so are its derivatives. There are more sophisticated versions of this; probably most notably Shi's estimates that I mention in the lecture and I may get around to in the future (its important to know if you are in the field and you should look it up, but I want to keep pushing on for now). Also important in the field are the Ecker-Huisken local estimates -- these papers are pretty computational but a good shorter project might be to present them and to try to ''explain'' the choices of test functions they make.

As I also mentioned in the lecture, a lot of very recent success has been had by using the Ricci flow to smooth metrics out (the point being to even understand metrics with unbounded curvature), here is a really nice survey by Topping (a leader in the Ricci flow) that you might find inspirational. If you like it he's also written a lot of other expository articles.

In the second lecture I discussed the ''geometric'' Arzela-Ascoli theorem; this will be used a lot later on because compactness-contradiciton arguments are used often: the idea is you can prove some sort of estimate by assuming its false and in so doing get a ''bad'' sequence of flows (although the technique is very general) and then apply compactness to take a limit. The limit will be forced by the assumptions to exhibit some rigidity behavior typically, which yields a contradiction. We used it in the second lecture to show that a smooth mean curvature flow on a time interval [0,T) can always be extended as long as the second fundamental form is bounded along the flow. First you use Arzela-Ascoli to extend the flow (as a map from M x [0,T)) to time T, and then you run your short time existence. The reason this is so important morally is because if you are using a flow, you need to have a precise idea why things might go bad. How exactly they go bad (i.e. singularity analysis) is a different story though and much more complicated.

Week 4:
This week there are three lectures. In the first lecture, I derved the mean curvature flow equation for graphs and then showed the comparison principle, that two disjoint mean curvature flows must stay disjoint at least when one of them is compact -- something I wished I made a bigger point of was that this is definitely a codimension 1 phenomenon unless you are in some special situation. For example, two linked round S^1 in R^3 will flow ``through'' each other under in the mean curvature flow. As a consequence, the study of singularities is essentially unavoidable for further understanding of the flow.

With that in mind, the next two lectures are about blowup analysis and we discuss the all important Huisken monotonicity formula and use it to show that tangent flows are self shrinkers under a type 1 curvature blowup rate hypothesis. I made a particular point on discussing what studying the singularity in renormalized coordinates had to do with the arguably more natural parabolic reescaling and reflected on the fact that there are ancient flows (with bounded Gaussian area, of course) which aren't self shrinkers - the fact the ancient limiting flow indeed comes from a blowup is key. Along the way I discussed self shrinkers and discussed how several different ways to define them (the self shrinker equation, a flow moving by certain dilations, minimal surface for Gaussian metric) were equivalent.

The sources I mentioned that I hadn't already linked to above are Huisken's paper and Ilmanen's preprint. Huisken's paper here is only 16 pages (and we only talked about 8 of them) and is easy to read, so I encourage everyone to take a look. Mantegazza has more comments and fills in more details, he puts a lot of effort in putting what Huisken did in the context of general backwards heat kernels so if you are more interested in analysis you might like to take a look at chapter 3 in his book.

For surfaces, it turns out the type 1 hypothesis is unnecessary to still get the limiting self shrinker is smooth, as shown by Ilmanen in 1995 preprint. Its amazing how important it is despite it never being published (as far as I know, its correct math) and is definitely worth taking a look at if you think you might work on geometric flows someday.

Weeks 5,6:
Week 5 we took a break because a lot of us had some extra responsibilities then and I wanted to let some of you catch up. Because this semester is very unusual I think its fair to go a little slower. For week 6, I discussed some general philosophy about compactness-contradiciton arguments and gave White's proof of the Brakke regularity theorem following his annals paper linked here. Its an easy to read paper making it great place to pick up this technique, which is good to know even if you aren't interested in geometric flows. The natural compactness theorem for us (since we are trying to live as much as possible in the category of smooth manifolds for this class) is Arzela-Ascoli, but of course there are ``intrinisic'' analogues as well. Probably the most well known analogous theorem in the more general setting is (Hamilton, Cheeger)-Gromov's compactness theorem.

Hamilton's version is for the Ricci flow, and you'll note in the statement you need so-called volume noncollapsing. The main motivation for such a theorem is to take blowup limits along the Ricci flow to study its singularities (which you would want to do to carry out Ricci flow with surgery), so do to so you want volume noncollapsing in this blowup sequence you take. This is a apriori a nontrivial assumption and one of the major breakthroughs of Perelman was to show this would always hold at least in the context of solving the 3D Poincare and geometrization conjectures via the Ricci flow.

Weeks 7.8:
I loosely presented the proof of Grayson's theorem via blowup analysis ala White (as far as I know), inspired largely by Chodosh's notes for White's class I mentioned above (in particular see sections 6,7). The idea is to start from the fact that the only self shrinking curves in the plane, due to Abresch and Langer and independently Epstein and Weinstein, are round circles and lines. The proof of this isn't so bad but in the lecture for the sake of brevity I showed that the geodesic curvature on such examples musn't switch signs, which should hopefully convince you that the statement is at least pretty plausible. A proof of the classification theorem can be found here (just a couple of pages).

The story isn't over with just knowing the classification of self shrinkers for the CSF in the plane though, one needs to rule out higher multiplicity convergence to a line and to do that you want to prove a so-called sheeting theorem which says in such cases the surface could be wrtten locally as some number of graphs over the limit - compare this in higher dimensions to White's sheeting theorem for mean convex MCF in this paper of his (by now you can probably see his importance to the field). With the sheeting theorem in hand, using more or less the same ideas used in the proof of the avoidance principle higher multiplicity convergence can then be ruled out.

I also talked a bit about how Grayson's theorem can be used to give a proof of Smale's theorem (maybe its better to say one of his theorems), this time refering to the fact that Diff(S^2) deformation retracts onto O(3), the group of rotations. The reduction of this to the statement ``the space of embedded curves in the plane is contractible'' follows from the appendix of Hatcher's proof of Smale's conjecture, which is the corresponding statement for Diff(S^3) (in even bigger dimensions, this statement is probably false I think).

There's a good chance you'll find Hatcher's paper to be a challenging read, partly because he takes a hands on approach and there are many cases to consider. That's one of the great things about flows: they do all the work for you! For that reason arguments using flows to me often seem to be easier to communicate and understand at a high level (even though there are often technical difficulties to overcome - consider that just establishing short time existence of the MCF realy rests on quite a bit of PDE). Recently, there was a Ricci flow proof of Smale's conjecture (and quite a bit more) by Bamler and Kleiner (its spread over a number of papers so i'll let you google it). It seems plausible (using the term loosely) to approach the problem by MCF though using the formulation that the space of embedded spheres is contractible, but this hasn't been accomplished yet and still seems to be a far way off. Generic MCF in R^3 (a topic I'll perhaps touch on later) has been pushed pretty far, but since one needs to consider families of surfaces there are additional complications.

The following don't really apply to our course but I include them to be on the safe side:
Academic integrity:
The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of students and/or the chairman of the Ethics Board beforehand. Read the “Statement on Ethics” at the Ethics Board website for more information.

Disability accommodations policy:
Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.