# Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics itself (at least when I try to search for "mean curvature flow applications", everything I find is phyics related or something like that), while, say, the Ricci flow, besides being interesting by itself (like the mean curvature flow), eventually provides a classification of all closed 3-manifolds (even in a lower level, you can use it to prove the sphere theorem or the uniformization theorem in dimension 2).

My intuition tells me that the mean curvature flow, just like any other extrinsic curvature flow, can't ever be used to get topological results about our manifolds, for example. Is this correct? If not, I'd really like seeing some interesting applications like this for the mean curvature flow (or other extrinsic flows).

I don't know if this is too elementary for mathoverflow but I decided to risk it since I didn't really find any questions like this on MSE.

EDIT: To make things a little bit more objective, what I really wanna see are topological results obtained from mean curvature flow. Naturally I think results using mean curvature flow can only be as strong as results that relate topology and mean curvature, and as far as I know, those are kinda pretty weak.

• "mean curvature flow, just like any other extrinsic curvature flow, can't ever be used to get topological results about our manifolds, for example. Is this correct?" What makes you think this? How could one show that something "can't ever be used"? For example, you could attempt to prove the nearby Lagrangian conjecture by using Lagrangian mean curvature flow to isotope an exact Lagrangian in a cotangent bundle to the zero section. This would show (a fortiori) that nearby Lagrangians are diffeomorphic to the zero-section. – Jonny Evans Nov 27 '19 at 22:03
• Of course, proving the nearby Lagrangian conjecture that way is currently out of reach and may not even work, but I just wanted to illustrate that there's no a priori reason that you can't prove topological results using mean curvature flow. The Thomas-Yau/Joyce conjectures (on special Lagrangians and Bridgeland stability conditions for Fukaya categories) have a similar flavour and the expectation seems to be that a better understanding of mean curvature flow with surgeries should help resolve these conjectures. – Jonny Evans Nov 27 '19 at 22:06
• @JonnyEvans My reasoning is that the mean curvature flow doesn't care about topology, only about mean curvature, and surfaces can be locally isometric (so that they have the same mean curvature) yet not homeomorphic. While sectional curvature, Ricci curvature and scalar curvature are all intrinsic, so in my head it would be much easier to obtain topological information from there rather than something that depends on ambient space. Of course, I don't trust my intuition and don't know whether this is right. So I'd really like to see some topological results using MCF. – Matheus Andrade Nov 28 '19 at 2:39
• The following isn't about MCF (so comment rather than an answer, sorry!), but is an example where an 'extrinsic' geometric flow has topological applications. A Kaehler manifold X can't have the same \pi_1 as a hyperbolic manifold M of dimension bigger than 2. Proof: if it did, apply harmonic map flow to the map X -> M which classifies the fundamental group; as M is negatively curved, you get a harmonic map, which then has to have 2-dimensional image by a rank estimate. This argument is due to Carlson and Toledo, building on the Eells-Sampson theory of harmonic maps and associated flow. – Jonny Evans Nov 28 '19 at 9:31
• – Terry Tao Nov 28 '19 at 17:35

Here is a list of some topological and geometric applications:

1. Huisken-Ilmanen used inverse MCF to prove the Riemannian Penrose inequality: https://projecteuclid.org/euclid.jdg/1090349447

2. Huisken-Sinestrari used MCF with surgery to classify two-convex hypersurfaces: https://link.springer.com/article/10.1007/s00222-008-0148-4

3. Buzano, Hershkovits and I used MCF with surgery to prove that the moduli space of two-convex embedded spheres is connected: https://arxiv.org/abs/1607.05604

4. Ketover and I used MCF with surgery to prove existence of smooth mean convex foliations and minimal spheres: https://arxiv.org/abs/1708.06567

5. Bernstein-Wang used MCF to classify hypersurfaces with low entropy: https://arxiv.org/abs/1511.00387

6. Ilmanen-White used MCF to prove sharp lower bounds on the density of minimal cones: https://arxiv.org/abs/1010.5068

7. Schulze proved the optimal isoperimetric inequality for surfaces in any codimension in Cartan-Hadamard manifolds: https://arxiv.org/abs/1802.00226

• Great answer! Thanks! – Matheus Andrade Nov 28 '19 at 18:35

Mu-Tao Wang (Math. Res. Lett. 2001) showed that any diffeomorphism $$f:S^2\to S^2$$ is isotopic to an isometry, which was originally shown by Smale (Proc. AMS 1959)

Mao-Pei Tsui and Mu-Tao Wang (Comm. Pure Appl. Math. 2004) showed that if $$f:S^n\to S^m$$ is area-decreasing on two-dimensional submanifolds, then $$f$$ is null-homotopic. (Gromov had shown this in the weaker context that the two-dimensional area distortion factor is sufficiently close to 0. Larry Guth (Geom. Func. Anal. 2013) has counterexamples if "two" is replaced by "three".)

Ivana Medos and Mu-Tao Wang (J Diff. Geom. 2011) showed that if $$f:\mathbb{CP}^n\to\mathbb{CP}^n$$ is a symplectomorphism such that $$f$$ and $$f^{-1}$$ have Lipschitz constants sufficiently close to one, then $$f$$ is symplectically isotopic to an isometry. (Gromov (Invent. Math. 1985) showed that in the case $$n=2$$ this is true without a condition on the Lipschitz factors.)

The method in each case is to deform the graph of $$f$$ by the mean curvature flow and to show a long-time existence and convergence result. So it is mean curvature in codimension larger than one, as opposed to most research in MCF.

As for mean curvature flow in codimension one, any smooth compact four-manifold which is homeomorphic to $$S^4$$ can be smoothly embedded in $$\mathbb{R}^5$$, and I think some people hope that a sufficiently good understanding of its mean curvature flow could prove the four-dimensional smooth Poincaré conjecture, roughly analogously to the Hamilton-Perelman proof of the three-dimensional Poincaré conjecture using Ricci flow. But it would probably be much more complicated, for analytic reasons.

I think there are two issues at play.

1. MCF of $$n$$ dimensional hypersurfaces is analogous to $$2n$$ dimensional Ricci flow (at least for $$n=1$$ and $$n=2$$).
2. The Riemann curvature tensor is more algebraically interesting than the second fundamental form so there are more possible interesting curvature conditions preserved.

For 1, there are several heuristic reasons to compare mean curvature flow of $$n$$-dimensional hypersurfaces to Ricci flow of $$2n$$ dimensional Riemannian manifolds. For instance, 1D MCF (i.e., curve shortening flow) is "trivial" in an analogous way that 2d Ricci flow is. Similarly, 2D MCF is "complex" (i.e. has many potential singularity models) in the same way 4D Ricci flow is and is beyond our present understanding. Ricci flow on three-manifolds is somewhere in between and so it was possible, with a lot of insight and work, to get results. However, there is no analog for MCF for dimensional reasons.

For the second point, there are nice applications of MCF to the study of hypersurfaces with various curvature conditions (e.g., two-convex). However, these are more exotic than curvature conditions that behave well with Ricci flow.

• I see, thanks for the answer! I'll still wait for other people's inputs before accepting yours though. Do you know of any topological results we can obtain using mean curvature flow? – Matheus Andrade Nov 28 '19 at 2:45
• If we knew more about MCF of two spheres (I think at this point we just need Ilmanen's multiplicity one conjecture) one could use it to show the space of embedded two spheres in $\mathbb{R}^3$ is contractible. This would give a new proof of Smale's conjecture that the group of diffeomorphisms on S^3 is homotopty equivalent to $O(3)$. – RBega2 Nov 28 '19 at 12:49