I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = (f(u), g(u)\text{cos }v, g(u)\text{sin }v),$$ $u \in [0, 1]$, $v \in [0, 2\pi]$. For example, what happens if $S$ is positively curved at all interior points? Or if $S$ is negatively curved at all interior points? Will it remain embedded for short time? Long time? How will it look in long time? I am trying to get some heuristic ideas. Thanks!
Note: Re-asked from MathOverflow.
The mean curvature flow (MCF) equation $$\frac{\partial F}{\partial t} = \vec H$$ is a (dengenerate) parabolic equation as under a local coordiantes we have $\vec H = \Delta_S F$.
So, in the case when $S$ is noncompact, it is not known if mean curvature flow has a short time solution. When $S$ is a surface of revolution, the MCF equation becomes a ODE on $f, g$. In that cases a solution might exists. I will check that if I have time later.
When $S$ is compact, the MCF always has a short time unique solution. It MUST becomes singular in finite time. Indeed, by checking $$\begin{split} \partial_t|F|^2 &= 2F \cdot \partial_t F \\ &=2 F\cdot \Delta_S F \\ &= \Delta_S |F|^2 - 2|\nabla_S F|^2 \\ &= \Delta_S |F|^2 - 4 \end{split}$$ Thus $G = |F|^2 +4t$ satisfies $\partial _t G = \Delta G$ and the maximum principle implies that $0\le |F_t|^2 \le \sup |F|^2 - 4t$ and so $t$ cannot be large.
Embedded surface stays embedded. This is proved by maximum principle. (Caution, this is not true if you consider, for example, mean curvature flow of a surface in $\mathbb R^{n\ge 4}$)
If $S$ is positively curved, then it is convex. A convex surface stays convex (By maximum principle on tensor). Huisken (84, Journal of differential geoemtry) shows that under the MCF a convex surfaces shrinks to a point $q$ in finite time, and, after a suitable rescaling, the rescale surface converges smoothly to the sphere centered at $q$.
You can find all the above facts in "Lectures on mean curvature flow" by X.P. Zhu.