It is well known that
Grayson's dumbbell neck-pinch<sup>1,2</sup> separates
into disconnected pieces under
mean curvature flow:
<hr />
![GraysonDumbells][1]
<br />
<sup>
[Image source: Simplicial Ricci Flow](http://inspirehep.net/record/1249871/plots).
(For contrast, see the earlier MO question,
[Intuition behind the Ricci flow](http://mathoverflow.net/a/143146/6094).)
</sup>
<hr />
Intuitively, it seems there might be another route to
morph any genus-zero surface embedded in $\mathbb{R}^3$ to a round sphere,
via "inflation."
Imagine slowly pumping air into the surface,
attempting to inflate it to a sphere.
Treat the surface as elastic/stretchable,
but do not allow the surface to pass through itself—it should remain embedded.
This would certainly work for the dumbbell,
but might get stuck for a pretzel-twisted surface.
I wonder if rendering the surface
"slippery"—zero surface-to-surface friction—would prevent
it from getting stuck.
> ***Q***. Has some notion of inflating a surface
(analogous to mean-curvature flow shrinking) been explored?
And perhaps found wanting?
I realize this question is not formalized, but I suspect the experts
can answer despite its vagueness.
<hr />
<sup>1</sup>M. A. Grayson, "A short note on the evolution of a surface by its mean curvature," *Duke Math. J.* 58 (3) (1989) 555–558.
([Euclid link](https://projecteuclid.org/euclid.dmj/1077307667).)
<br />
<sup>2</sup>Tobias Holck Colding, William P. Minicozzi II and Erik Kjær Pedersen.
"Mean curvature flow."
*Bull. Amer. Math. Soc.* 52 (2015), 297-333.
([AMS link](http://www.ams.org/journals/bull/2015-52-02/S0273-0979-2015-01468-0/).)
[1]: https://i.sstatic.net/b3Yts.jpg