THE GRADUATE STUDENT SECTION

WHAT IS…

a GeneralisedMean-Curvature Flow?

Hui YuCommunicated by Cesar E. Silva

To untie a shoelace by pulling both ends of the string,shrinking the bunny ears until they disappear, gives greatintellectual pleasure. Mathematically, we unknotted anunknot by decreasing the length of the curve containedin the knot. This procedure turns out to be a rathereffective way to simplify geometries. It is one of thereasons why geometers like Richard Hamilton study theso-called curve-shortening flow, where an initial curve iscontinuously deformed so that its length decreases overtime.

The mean-curvature flowenjoys niceproperties.

The way to do this isto move or flow the curvenormal to itself at a rate pro-portional to its curvature.There is a similar flow as inFigure 1 to reduce the areaof a (𝑑 − 1)-dimensionalsurface in ℝ𝑑, at a rateproportional to the meancurvature, the mean of the

curvatures of 1-dimensional slices by orthogonal planes.Apart from shortening curves, themean-curvature flow

enjoys other nice properties, such as locality, since themotion of a point depends only on an infinitesimal neigh-bourhood, and rigid-motion invariance, since “length” isinvariant under rigid-motions. Other properties are more

Hui Yu is a graduate student at UT–Austin and will be Ritt Assis-tant Professor at Columbia University starting in August.Yu is partially supported by NSF grant DMS-1500871.His e-mail address is hyu@math.utexas.edu.For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1542

Figure 1. The surface of a dumbbell develops asingularity before turning into two disconnectedspherical shapes.

subtle. If you know that the mean curvature is the Lapla-cian of the function that measures distances from pointsto themanifold, then you can see that themotion is drivenby a parabolic equation. This parabolicity gives rise tothe inclusion principle, which says that if we deform twodomains 𝑈 ⊂ 𝑉 by letting their respective boundariesfollow the mean-curvature flow, then these domains re-main ordered. This is a consequence of the comparisonprinciple for parabolic equations. It is also a consequenceof geometry: If 𝑉 failed to contain 𝑈 at some time, thenthere would be a critical moment when 𝑈 is still inside 𝑉but their boundaries touch at a certain point. Doodle a bitwith curves, and you see that in this situation, to reduce“length,” the boundaries are instantly pulled away fromeach other. Conclusion: 𝑈 remains inside 𝑉.

All these nice properties do not make the mean-curvature flow perfect. Onemajor flaw is that it is not welldefined globally in time. After convex surfaces shrink intopoints, the flow is no longer defined. But this example doesnot capture the worst behaviour, because the object itselfdisappears after the critical point. Think of a dumbbell, asin Figure 1. If the plates are too big compared to the neck,the dumbbell turns into two smaller plates connected bya one-dimensional “string.” Again the flow ceases to bewell defined afterwards.

Butwhatwehave calledaflaw is actually something thatmakes the mean-curvature flow analytically interesting.

580 Notices of the AMS Volume 64, Number 6

THE GRADUATE STUDENT SECTIONWe can look for generalised/weak versions of the flow thatcontinue the motion after the critical time. This searchfor generalised mean-curvature flows has become one ofthe central themes in analysis in the past few decadesand leads to a lot of exciting mathematics.

Amanifold is the zero set of the function thatmeasuresdistances between any point and the manifold. Thisdistance function satisfies a parabolic equation in themean-curvature flow. Therefore, one natural way to get ageneralised mean-curvature flow is to study generalisedsolutions to such equations, and then define their zerolevel sets to be solutions to the generalised flow. This so-called level-set method, as described in theDecember 2016Notices cover story by Colding and Minicozzi, reduces thenotion of a weak geometric flow to the notion of a weaksolution to parabolic equations. Luckily, the viscositysolution provides exactly what we need. In fact, muchof the theory of viscosity solutions is inspired by thisapproach to a generalised mean-curvature flow.

The search forgeneralised

mean-curvature

flows leads to alot of excitingmathematics.

Another approach takesadvantage of the inclusionprinciple. Suppose our mani-fold is the boundary of somedomain 𝑈. Then the inclu-sion principle dictates thatany domain initially contain-ing 𝑈 continues to do so inthe future. Although 𝑈 mightbe a very rough set where thesmooth mean-curvature flowdoes not exist, we can alwaysfind nice sets containing 𝑈.Starting from these nice setswe do have flows, and wemight simply define a generalised mean-curvature flowof 𝑈 to be the intersection of all flows starting fromthese nice sets. This method of minimal barriers mightremind the reader of the method of Perron, wherebyone constructs solutions to elliptic equations as leastsupersolutions.

A third approach, which goes back to the curve-shortening property, begins by defining a weak notion ofthe length of curves or the area of surfaces. The hopeis that, using this generalised length/area, we would beable to consider the flow for much rougher objects. Suchgeneralised length/area is provided by the theory of setsof finite perimeter from geometric measure theory. Ateach instant of time, we look for a set of finite perimeterthat most efficiently reduces this generalised length/areaand define this set to be the solution to the generalisedflow. This gives rise to the notion of flat flows.

A related approach follows from the observation thatthe length/area can be seen as the limit of the Ginzburg-Landau energy functional when a certain parameter 𝜖goes to 0. For each positive 𝜖, we have a globally well-defined gradient flow for this energy. We simply defineour generalisedmean-curvature flow as the limit as 𝜖 goesto zero of this gradient flow. This is called the phase fieldapproach.

The literature on generalised mean-curvature flows isparticularly rich, containing many methods beyond thescope of a two-page article. To earn the label “generalisedmean-curvature,” however, a flow needs to satisfy severalrequirements: firstly, it should be globally defined; sec-ondly, it should be consistent with smooth flows as longas the latter exist; and lastly, it should inherit certain keyfeatures of the smooth flows.

These naive-looking requirements sometimes lead tochallenging problems that have inspired a great dealof research. Because generalised mean-curvature flowsprovide a paradigm for the study of all kinds of geometricflows, results in this area are of especially wide interest.

References[1] G. Bellettini, Lecture notes on mean curvature flow, bar-

riers and singular perturbations, Publications of the ScuolaNormale Superiore, Scuola Normale Superiore 12 (2013).

[2] H. Yu, Motion of sets by curvature and capacity potential, sub-mitted. Available at https://arxiv.org/abs/1701.02837.

Photo CreditsImage in Figure 1 by Oliver Knill, Harvard University.Photo of Hui Yu by Yunan Yang, PhD student at UT Austin.

June/July 2017 Notices of the AMS 581