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It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:

         GraysonDumbells
          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.

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.

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.)
2Tobias Holck Colding, William P. Minicozzi II and Erik Kjær Pedersen. "Mean curvature flow." Bull. Amer. Math. Soc. 52 (2015), 297-333. (AMS link.)

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    $\begingroup$ Huisken--Ilmanen's (weak) inverse mean curvature flow projecteuclid.org/euclid.jdg/1090349447 preserves embeddedness, and the surface will remain connected as long as it starts connected, and it will become a (giant) round sphere for large time. It is unlikely to be a smooth flow for all time, but instead there should be some times when the surface jumps outwards... $\endgroup$
    – Otis Chodosh
    Commented May 4, 2015 at 1:04
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    $\begingroup$ The problem with the wIMCF is that the handle of the dumbbell will get immediately inflated to the cylinder the same width as the heads of the dumbbell. If you are happy with that then great. But as Otis noted, the definition of the wIMCF allows this kinds of jumps which is not very "flowy". $\endgroup$
    – Willie Wong
    Commented May 4, 2015 at 12:08
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    $\begingroup$ Hmm... I was being slightly imprecise in my previous comment. So lest you be misled: what I said about the inflation is more-or-less true (the width of the cylinder will be similar to, but not exactly the same as, the width of the head) if you assume the handle is sufficiently short and skinny. // Perhaps a better illustration of the problem is that the wIMCF allows changes in topology in its definition: if you start with two spheres, or a skinny torus, both will eventually become spheres under this flow. $\endgroup$
    – Willie Wong
    Commented May 4, 2015 at 12:47
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    $\begingroup$ @JosephO'Rourke: Did you mean to discuss Ricci flow? I was confused by the title of your question, since you actually seem to be looking for something related to mean curvature flow. $\endgroup$
    – Ian Agol
    Commented May 4, 2015 at 16:13
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    $\begingroup$ You might be interested in this cs.jhu.edu/~misha/MyPapers/SGP12.pdf paper which introduces a modification of the mean curvature flow (motivated heuristically by numerical analysis considerations). The authors claim that numerical evidence suggests that it doesn't form neckpinch singularities. However, this is not rigorously shown and the geometric meaning of the flow is a bit unclear. The article does have some pretty pictures. $\endgroup$
    – Rbega
    Commented May 4, 2015 at 16:54

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My question was usefully answered in the comments: Otis Chodosh pointed me to the inverse mean-curvature flow as described, e.g., in this paper:

Huisken, Gerhard, and Tom Ilmanen. "The inverse mean curvature flow and the Riemannian Penrose inequality." Journal of Differential Geometry. 59.3 (2001): 353-437.

and Willie Wong explained how it would evolve dumbbells. Rbega pointed me to this paper

Kazhdan, Michael, Jake Solomon, and Mirela Ben‐Chen. "Can Mean‐Curvature Flow be Modified to be Non‐singular?." Computer Graphics Forum. Vol. 31. No. 5. Blackwell Publishing Ltd, 2012.

which contains this image of their modified "conformalized mean-curvature flow" applied to dumbbells:

  Fig3Dumbbels

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