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?