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

 

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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