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## Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

26 November, 2018 in math.AP, math.MG, paper | Tags: bilipschitz embedding, Heisenberg group, Nash embedding theorem | by Terence Tao | 11 comments

I’ve just uploaded to the arXiv my paper “Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion“, submitted to Revista Matematica Iberoamericana. This paper concerns the extent to which one can accurately embed the metric structure of the Heisenberg group

into Euclidean space, which we can write as with the notation

Here we give the right-invariant Carnot-Carathéodory metric coming from the right-invariant vector fields

but not from the commutator vector field

This gives the geometry of a Carnot group. As observed by Semmes, it follows from the Carnot group differentiation theory of Pansu that there is no bilipschitz map from to any Euclidean space or even to , since such a map must be differentiable almost everywhere in the sense of Carnot groups, which in particular shows that the derivative map annihilate almost everywhere, which is incompatible with being bilipschitz.

On the other hand, if one *snowflakes* the Heisenberg group by replacing the metric with for some , then it follows from the general theory of Assouad on embedding snowflaked metrics of doubling spaces that may be embedded in a bilipschitz fashion into , or even to for some depending on .

Of course, the distortion of this bilipschitz embedding must degenerate in the limit . From the work of Austin-Naor-Tessera and Naor-Neiman it follows that may be embedded into with a distortion of , but no better. The Naor-Neiman paper also embeds into a finite-dimensional space with independent of , but at the cost of worsening the distortion to . They then posed the question of whether this worsening of the distortion is necessary.

The main result of this paper answers this question in the negative:

Theorem 1There exists an absolute constant such that may be embedded into in a bilipschitz fashion with distortion for any .

To motivate the proof of this theorem, let us first present a bilipschitz map from the snowflaked line (with being the usual metric on ) into complex Hilbert space . The map is given explicitly as a Weierstrass type function

where for each , is the function

and are an orthonormal basis for . The subtracting of the constant is purely in order to make the sum convergent as . If are such that for some integer , one can easily check the bounds

with the lower bound

at which point one finds that

as desired.

The key here was that each function oscillated at a different spatial scale , and the functions were all orthogonal to each other (so that the upper bound involved a factor of rather than ). One can replicate this example for the Heisenberg group without much difficulty. Indeed, if we let be the discrete Heisenberg group, then the nilmanifold is a three-dimensional smooth compact manifold; thus, by the Whitney embedding theorem, it smoothly embeds into . This gives a smooth immersion which is -automorphic in the sense that for all and . If one then defines to be the function

where is the scaling map

then one can repeat the previous arguments to obtain the required bilipschitz bounds

for the function

To adapt this construction to bounded dimension, the main obstruction was the requirement that the took values in orthogonal subspaces. But if one works things out carefully, it is enough to require the weaker orthogonality requirement

for all , where is the bilinear form

One can then try to construct the for bounded dimension by an iterative argument. After some standard reductions, the problem becomes this (roughly speaking): given a smooth, slowly varying function whose derivatives obey certain quantitative upper and lower bounds, construct a smooth oscillating function , whose derivatives also obey certain quantitative upper and lower bounds, which obey the equation

We view this as an underdetermined system of differential equations for (two equations in unknowns; after some reductions, our can be taken to be the explicit value ). The trivial solution to this equation will be inadmissible for our purposes due to the lower bounds we will require on (in order to obtain the quantitative immersion property mentioned previously, as well as for a stronger “freeness” property that is needed to close the iteration). Because this construction will need to be iterated, it will be essential that the regularity control on is the same as that on ; one cannot afford to “lose derivatives” when passing from to .

This problem has some formal similarities with the isometric embedding problem (discussed for instance in this previous post), which can be viewed as the problem of solving an equation of the form , where is a Riemannian manifold and is the bilinear form

The isometric embedding problem also has the key obstacle that naive attempts to solve the equation iteratively can lead to an undesirable “loss of derivatives” that prevents one from iterating indefinitely. This obstacle was famously resolved by the Nash-Moser iteration scheme in which one alternates between perturbatively adjusting an approximate solution to improve the residual error term, and mollifying the resulting perturbation to counteract the loss of derivatives. The current equation (1) differs in some key respects from the isometric embedding equation , in particular being linear in the unknown field rather than quadratic; nevertheless the key obstacle is the same, namely that naive attempts to solve either equation lose derivatives. Our approach to solving (1) was inspired by the Nash-Moser scheme; in retrospect, I also found similarities with Uchiyama’s constructive proof of the Fefferman-Stein decomposition theorem, discussed in this previous post (and in this recent one).

To motivate this iteration, we first express using the product rule in a form that does not place derivatives directly on the unknown :

This reveals that one can construct solutions to (1) by solving the system of equations

for . Because this system is zeroth order in , this can easily be done by linear algebra (even in the presence of a forcing term ) if one imposes a “freeness” condition (analogous to the notion of a free embedding in the isometric embedding problem) that are linearly independent at each point , which (together with some other technical conditions of a similar nature) one then adds to the list of upper and lower bounds required on (with a related bound then imposed on , in order to close the iteration). However, as mentioned previously, there is a “loss of derivatives” problem with this construction: due to the presence of the differential operators in (3), a solution constructed by this method can only be expected to have two degrees less regularity than at best, which makes this construction unsuitable for iteration.

To get around this obstacle (which also prominently appears when solving (linearisations of) the isometric embedding equation ), we instead first construct a smooth, low-frequency solution to a low-frequency equation

where is a mollification of (of Littlewood-Paley type) applied at a small spatial scale for some , and then gradually relax the frequency cutoff to deform this low frequency solution to a solution of the actual equation (1).

We will construct the low-frequency solution rather explicitly, using the Whitney embedding theorem to construct an initial oscillating map into a very low dimensional space , composing it with a Veronese type embedding into a slightly larger dimensional space to obtain a required “freeness” property, and then composing further with a slowly varying isometry depending on and constructed by a quantitative topological lemma (relying ultimately on the vanishing of the first few homotopy groups of high-dimensional spheres), in order to obtain the required orthogonality (4). (This sort of “quantitative null-homotopy” was first proposed by Gromov, with some recent progress on optimal bounds by Chambers-Manin-Weinberger and by Chambers-Dotterer-Manin-Weinberger, but we will not need these more advanced results here, as one can rely on the classical qualitative vanishing for together with a compactness argument to obtain (ineffective) quantitative bounds, which suffice for this application).

To perform the deformation of into , we must solve what is essentially the linearised equation

of (1) when , (viewed as low frequency functions) are both being deformed at some rates (which should be viewed as high frequency functions). To avoid losing derivatives, the magnitude of the deformation in should not be significantly greater than the magnitude of the deformation in , when measured in the same function space norms.

As before, if one directly solves the difference equation (5) using a naive application of (2) with treated as a forcing term, one will lose at least one derivative of regularity when passing from to . However, observe that (2) (and the symmetry ) can be used to obtain the identity

and then one can solve (5) by solving the system of equations

for . The key point here is that this system is zeroth order in both and , so one can solve this system without losing any derivatives when passing from to ; compare this situation with that of the superficially similar system

that one would obtain from naively linearising (3) without exploiting the symmetry of . There is still however one residual “loss of derivatives” problem arising from the presence of a differential operator on the term, which prevents one from directly evolving this iteration scheme in time without losing regularity in . It is here that we borrow the final key idea of the Nash-Moser scheme, which is to replace by a mollified version of itself (where the projection depends on the time parameter). This creates an error term in (5), but it turns out that this error term is quite small and smooth (being a “high-high paraproduct” of and , it ends up being far more regular than either or , even with the presence of the derivatives) and can be iterated away provided that the initial frequency cutoff is large and the function has a fairly high (but finite) amount of regularity (we will eventually use the Hölder space on the Heisenberg group to measure this).

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