It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:
![GraysonDumbells][1]
[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).)
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 Ricci-flow experts can answer despite its vagueness.
1M. 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).)
2Tobias 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/).)