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:
<hr />
&nbsp;
![Fig3Dumbbels][1]
<hr />


  [1]: https://i.sstatic.net/QG4Pr.png