It is well known that
Grayson's dumbbell neck-pinch^{1,2} separates
into disconnected pieces under
mean curvature flow:

^{ Image source: Simplicial Ricci Flow. (For contrast, see the earlier MO question, Intuition behind the Ricci flow.) }

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.

. Has some notion of inflating a surface (analogous to mean-curvature flow shrinking) been explored? And perhaps found wanting?Q

I realize this question is not formalized, but I suspect the experts can answer despite its vagueness.

^{1}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.)

^{2}Tobias Holck Colding, William P. Minicozzi II and Erik Kjær Pedersen. "Mean curvature flow."

*Bull. Amer. Math. Soc.*52 (2015), 297-333. (AMS link.)