Have a look at what I've saved, if you like. Here's the link to the mathematical pages, if you like.
I've been out of the game for a while (as a spectator) and have forgotten the current state of affairs of the Isoperimetric Problem in the Heisenberg group. However, there's a new article in preparation with another partial solution; some of the abstract is omitted.
- Area-stationary surfaces in the Heisenberg group H^1 [link]
Authors: Manuel Ritoré, César Rosales - We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\'eodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero ..
.. we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points.
As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.
If the article is correct, then what remains is regularity of the boundary of the extremal set. Immediately I wonder why the C2-smoothness is used by the authors, other than as a convenience. With more work, can it be done without such an a priori assumption?
I should read the damn paper and be done with it .. if only I could convince myself that
mathematics ≠ thesis work.