## A contribution to the Metric embedding problem list

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 ${d_{Heis}}$ on ${\mathbb{R}^3}$, 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 ${d_{Eucl}}$. The converse estimate

$\displaystyle \begin{array}{rcl} d_{Heis}\leq d_{Eucl}+d_{Eucl}^{1/2} \end{array}$

is not hard. Gromov’s question is wether, with a change of coordinates, this inequality can be substantially improved. Precisely,

Does there exist ${\alpha>\frac{1}{2}}$, a constant ${C>0}$ and a local homeomorphism ${\phi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}}$ such that for all ${p}$, ${q\in\mathbb{R}^3}$, close enough to the origin,

$\displaystyle \begin{array}{rcl} \frac{1}{C}\,d_{Eucl}(p,q)^{1/\alpha}\leq d_{Heis}(\phi(p),\phi(q))\leq C\,d_{Eucl}(p,q)^{\alpha}. \end{array}$

Removing the exponent ${1/\alpha}$ 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 ${d_{Heis}^{1-\epsilon}}$, ${\epsilon>0}$, admits a biLipschitz embedding into some Euclidean space ${\mathbb{R}^N}$. A recent result of Naor and Neiman even states that ${N}$ can be chosen independant on ${\epsilon}$. However, their ${N}$ is far from ${3}$. The requirement that the range Euclidean space is ${3}$-dimensional (in other words, that ${\phi}$ 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 ${\alpha}$ in above inequality for various distributions, see [G], but could never get sharp bounds. See [P] for an exposition.

References

[G] Mikhael Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhaüser, Basel, 1996.

[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