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](https://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