Metric differentiation, monotonicity and maps to \(L^{1}\)
The authors study Lipschitz embeddings in the category of topological spaces. As is well known, Lipschitz continuity is stronger than continuity, since, roughly speaking, it requires bounded variations. More precisely, if \(f:X\to Y\) is a mapping between metric spaces \((X,d_X)\) and \((Y,d_Y)\), then \(f\) is Lipschitz continuous iff there exists a constant \(K\geq 0\) such that for all \(x_1\not= x_2\), \(d_Y(f(x_1),f(x_2))/d_X(x_1,x_2)\leq K\). Furthermore, \(f\) is called bi-Lipschitz if there exists a \(K\geq 1\) such that \({{1}\over{K}}d_X(x_1,x_2)\leq d_Y(f(x_1),f(x_2))\leq K d_X(x_1,x_2)\) for all \(x_1\not= x_2\). A surjective bi-Lipschitz mapping is an isomorphism in the Lipschitz category of metric spaces (where morphisms are Lipschitz maps). For a bijective, bi-Lipschitz mapping \(f\), the inverse function is also bi-Lipschitz with the same constant \(K\) of \(f\). It is interesting to emphasize that a differentiable function \(f:{\mathbb R}\to{\mathbb R}\) is not necessarily Lipschitz. Conversely, a Lipschitz function \(g:{\mathbb R}\to{\mathbb R}\) is absolutely continuous and therefore is differentiable almost everywhere. Furthermore \(\|Df(x)\|\leq K\), where \(K\) is the Lipschitz constant of \(f\) (Rademacher's theorem). Rademacher's theorem extends to Lipschitz functions \(f:X\to{\mathbb R}\), where \((X,\mu)\) is a doubling measure metric space satisfying a Poincaré inequality (Cheeger's theorem). A metric space is doubling if there is a constant \(N\) such that any ball can be covered by at most \(N\) balls of half the radius. If \((X,d)\) is a doubling metric space and \(0<\alpha<1\), then there is a bi-Lipschitz embedding \(f:(X,d^\alpha)\to{\mathbb R}^n\) (Assouad's theorem). By the concept of bi-Lipschitz mappings, it is possible to recognize in the category of topological manifolds a little subcategory of Lipschitz manifolds, where the Lipschitz structure is defined by means of a pseudogroup of bi-Lipschitz homeomorphisms. A PL structure gives rise to a unique Lipschitz structure. This paper fits in a series of companion ones by the same authors, devoted to study Lipschitz maps from metric spaces to \(L^1\equiv L^1({\mathbb R},{\mathcal L})\), where \({\mathcal L}\) is the Lebesgue measure on \({\mathbb R}\). See also the paper by \textit{S. D. Pauls} [Commun. Anal. Geom. 9, No.~5, 951--982 (2001; Zbl 1005.53033)]. The starting point is the work by \textit{J. Cheeger} [Geom. Funct. Anal. 9, No.~3, 428--517 (1999; Zbl 0942.58018)], and a proof of Lee-Naor's following conjecture: \(({\mathbb H},d^{\mathbb H})\), the Heisenberg group equipped with its Carnot-Carathéodory metric, does not admit a bi-Lipschitz embedding into \(L^1\). This is a generalization of Pansu's theorem [1989] for bi-Lipschitz embeddings \({\mathbb H}\to {\mathbb R}^n\), for any \(n\). The authors emphasize the significance of this conjecture in the context of theoretical computer science. In particular, the authors refer to their joint paper [Ann. Math. (2) 171, No.~2, 1347--1385 (2010; Zbl 1194.22009)], presenting the details of results announced there, giving also an alternative proof of the main theorem (that just proves Lee-Naor's conjecture). The proof uses the metric differentiation theorem of S. Pauls [1999/2001], and the cut metric description, just considered by the same authors in the above quoted joint paper, in order to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group. They announce also that a quantitative version of this classification argument is used in a further forthcoming joint paper with \textit{A. Naor} [`Compression bounds for Lipschitz maps from the Heisenberg group to \(L^1\)', \url{arXiv:0910.2026} (2009)]. The paper, after a detailed introduction, where the principal results are resumed and compared with previous ones, splits into seven more sections: 2. Preliminaries. 3. Monotonicity and geodesic maps to \(L^1\). 4. The classification of precisely monotone sets. 5. The classification of monotone sets. 6. The proof of a weak version of Theorem 1.3. 7. Uniqueness of cut measures. 8. Cut measures which are standard on lines. Theorem 1.3. For almost every \(x\in {\mathbb H}\), there is a semi-norm \(\|.\|_x\) on \({\mathbb R}^2\) such that \(\rho_{x,\lambda}(z_1,z_2)\to\|\pi(z_1)-\pi(z_2)\|_x\), as \(\lambda\to 0\), uniformly on compact subsets of \({\mathbb H}\times{\mathbb H}\). Here, \(\pi:{\mathbb H}\times{\mathbb H}\to{\mathbb H}\times{\mathbb H}/[{\mathbb H},{\mathbb H}]\cong {\mathbb R}^2\) is the abelianization homomorphism. In particular, \(\rho_{x,\lambda}\) converges to a pseudodistance which is zero along fibers of \(\pi\), and hence \(f\) is not bi-Lipschitz in any neighborhood of \(x\). Here \({\mathbb H}\) denotes the Heisenberg group equipped with the Carnot-Carathéodory metric \(d\). Furthermore, \(\rho:{\mathbb H}\times{\mathbb H}\to[0,\infty)\) is defined by \(\rho(x_1,x_2)=\|f(x_1)-f(x_2)\|_{L^1}\), where \(f:{\mathbb H}\to L^1\) is a Lipschitz map. For \(\lambda\in(0,\infty)\), and \(x\in{\mathbb H}\), \(\rho_{x,\lambda}\) is defined by the following formula: \(\rho_{x,\lambda}(z_1,z_2)=\rho(xs_\lambda z_1,xs_\lambda z_2)/\lambda\), with \(s_\lambda:{\mathbb H}\to{\mathbb H}\) the automorphism which scales distances by the factor \(\lambda\). The weak version is Theorem 6.1. Let \(Z\) be the infinitesimal generator of Center\(({\mathbb H})\), so Center\(({\mathbb H})=\{\exp tZ\mid t\in{\mathbb R}\}\). If \(f:{\mathbb H}\to L^1\) is a Lipschitz map, then for a full measure set of points \(p\in{\mathbb H}\), \(\liminf_{t\to 0}{{d(f(p),f(p\exp tZ))}\over{d(p,p \exp tZ)}}=0\). This is enough to imply the nonexistence of bi-Lipschitz embeddings \({\mathbb H}\to L^1\), and its proof does not use harmonic analysis (that, instead, is considered in the next section to prove Theorem 1.3.).
global analysis in infinite-dimensional manifolds
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