a list of links, and a new preprint on the arXiv.

I've built a new webpage of bookmarks on del.icio.us recently, and many of them are tagged by the term "mathematics."

Have a look at what I've saved, if you like. Here's the link to the mathematical pages, if you like.

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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.

I might not understand the hypotheses of "area-stationary surfaces," but it seems plausible that one main difficulty of the Heisenberg Isoperimetric problem -- the symmetry -- may have been resolved. Once I get out of this holiday/post-prelim funk and summon enough will to set aside my thesis work, I'll have to read it carefully ..

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.