It is well known that if you start with a domain $\Omega \subset \mathbb{R}^d$ which is uniformly convex, then it converges exponentially fast to the ball when evolved under volume preserving mean curvature flow (VPMCF).

In $\mathbb{R}^2$, if one starts with a smooth simply connected domain, then after a finite amount of time it will become convex, and converge to the ball.

Question: If $\Omega \subset \mathbb{R}^d$ is smooth and connected, does it converge (perhaps after surgery) to a constant mean curvature (CMC) surface? Is it possible to classify the type of CMC surface it converges to based on the intial domain $\Omega$? I am hoping mostly for direction to the appropriate literature in this case, as mathscinet is flooded with papers on the subject.

Update: I would like to know in particular, on the torus $\mathbb{T}^2$, if I start with a set $\Omega \subset \mathbb{T}^2$ which satisfies $|\Omega|=1/2$, and I evolve it under VPMCF, does it eventually converge to the stripe pattern?


For manifolds those don't possess positive mean curvature, singularities are expected to occur in general. Unfortunately, we don't have a surgery theory in this case. The most successful result on surgery were due to Huisken and Sinestrari, which is concerning to some weak convexity (e.g. 2-convex).

More about surgery and convexity: In 3D Ricci flow(RF), the positivity of curvature of blow-up solution comes from Hamilton-Ivey pinching estimate. However, in 2D MCF(which corresponds to 3D RF), there is NO such pinching estimate. This is the reason that we need the convexity assumption to exclude some terrible models of singularities. (Huisken claimed that Sinestrari and he had classified 2D $\epsilon$-pinched ancient solution.) We should also notice that there is NO generic surgery for dimension $n\geq 4$ even for the RF.

  • $\begingroup$ By the way, you may try to consider the level set flow, which behaves better then MCF when passing the singularity. In this ansatz, the convexity is not as crucial as in MCF. $\endgroup$ Mar 9 '12 at 3:46
  • $\begingroup$ In two dimensions, do smooth connected curves eventually become convex under VPMCF as in in the regular MCF case? $\endgroup$
    – Dorian
    Mar 9 '12 at 11:31
  • $\begingroup$ Sorry, I think that I confused the VPMCF with normalized MCF. For VPMCF, I don't know any result related to your questions. A friend of mine studies VP elastic flow on the plane (for a segment, not a closed curve). But he never mentions to me any result related to your questions. $\endgroup$ Mar 12 '12 at 1:30

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