wow. everyone has preprints now.

today i thought i had proved something great.

feeling good, i decided to take a little break and procrastinate, so i searched the arXiv and then a few preprint servers ..

.. and found that an acquaintance of mine has a preprint up. he's the third author.

Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane [link]

Authors: Luigi Ambrosio - Bruce Kleiner - Enrico Le Donne

Abstract: We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they proved that, for almost every x, E has a unique tangent at x, and this tangent is a halfspace.

wow. awesome stuff.

it leads to all sorts of compatibility questions, in my mind.

there is a substantial theory of rectifiability on Carnot groups, these days, using explicit techniques from the sub-Riemannian geometric structure of such spaces. unlike some examples in the analysis on metric spaces, in sub-Riemannian geometry you are permitted vector fields and contact forms and explicit formulas for geodesics, in some cases.

put simply, you can work with pretty smooth things.

on the other hand, a friend of mine (a mathematical brother, actually [1]) has shown that there cannot be currents on Carnot groups in the sense of metric spaces (after Ambrosio and Kirchheim).

in contrast, on Euclidean spaces the theory of currents and rectifiability go hand in hand. it makes, at least for me, a confusing geometric measure theory.

at any rate, people i know seem to do such fine things. take these guys:

john's preprints on the arXiv, of which there are two.
kevin's preprints, also two in number.

.. and all my officemates are writing or finishing preprints of their own. the world is a competitive place, of course.

it's just that sometimes,

you don't realise how good the competition is.

[1] that is, we have the same advisor. speaking of which, he is soon to write up his own work, as well as marie, another mathematical sib.