I would like to advertise Gromov’s Hölder equivalence problem.
It is not quite an embedding problem, but sounds close. For me, the Heisenberg group is a metric on , defined by minimizing the Euclidean length of curves tangent to a certain plane distribution (i.e. curves satisfying dz=ydx). It is clearly larger than . The converse estimate
is not hard. Gromov’s question is wether, with a change of coordinates, this inequality can be substantially improved. Precisely,
Does there exist , a constant and a local homeomorphism such that for all , , close enough to the origin,
Removing the exponent on the left hand side may make the problem easier.
Since Heisenberg group is a doubling metric space, Assouad’s theorem asserts that every snowflaked metric , , admits a biLipschitz embedding into some Euclidean space . A recent result of Naor and Neiman even states that can be chosen independant on . However, their is far from . The requirement that the range Euclidean space is -dimensional (in other words, that is a homeomorphism) gives Gromov’s question a different flavour.
The above question generalizes to all bracket generating plane distributions. Gromov has put a lot of ingeniosity in getting upper bounds on the possible exponents in above inequality for various distributions, see [G], but could never get sharp bounds. See [P] for an exposition.
[P] Pierre Pansu, Submanifolds and differential forms in Carnot manifolds, after M. Gromov et M. Rumin. http://www.math.u-psud.fr/~pansu/liste-prepub.html