Thanks, this is a nice idea and it seems like something along these lines should work.

@Alexandre Eremenko, the topology is already defined as the space of continuous injective maps from the disk to the sphere modulo homeomorphisms of the disk. So two domains are close provided that they admit parameterizations which are $C^0$ close. This is different from Hausdorff topology, where as you say bad things happen.

@RBega, I don't think that CSF is going to work even for smooth curves, because if we take a pair of smaller circles on either side of a great circle and parallel to it they will float towards antipodal points.

@RBega, as far as I know curve shortening flow would require the curve to be rectifiable. Also I do not know how you would keep the area constant.

@Marco Golla, it does not work when the center of mass is at the origin.