Bi-Lipschitz arcs in metric spaces with controlled geometry (Q6620360)

Let \(f:A\to X\) be a map with \(A\subset Y\) with some nice property. When can such a map be extended to all of \(Y\) while preserving this property? Such extension problems are widely studied and the introduction presents their background well. This paper studies this problem when property is being a bi-Lipschitz map, \(Y=\mathbb{R}\) and \(X\) is a complete metric space. A map is bi-Lipschitz, if \(|x-y|\lesssim d(f(x),f(y))\lesssim |x-y|\). The main theorem, Theorem 1.2., shows that extension is possible if \(X\) is \(Q\)-Ahlfors regular and satisfies a \(p\)-Poincaré inequality for some \(1<p\leq Q-1\). The introduction explains this problem and some further results.\N\NWhere does the difficulty lie with this problem? The strategy is explained at the end of the introduction. First, without loss of generality, one can take \(A\) to be a closed set. Given this, one notices directly that \(\mathbb{R} \setminus A\) is a countable union of disjoint intervals. The task is to expand the definition to all these intervals while preserving the bi-Lipschitz condition. This is achieved by gluing in curves for each missing piece.\N\NThe bi-Lipschitz condition consists of two parts: an upper bound and a lower bound. The upper bound is the familiar Lipschitz condition, and it is easy to see what it requires. Lipschitz extendability is equivalent to \(X\) being quasiconvex. Notice Lemma 4.1. of Wenger and Lytchak which allows to convert a quasiconvex curve into a bi-Lipschitz curve in quasicnvex spaces. Thus, the issue is really the lower bound, which requires that the glued curves avoid each other and the image of \(f:A\to X\) sufficiently well. This is not always possible, as the examples on page 1889 of the paper show nicely. The issue is topological and quantitative: the map \(f:A\to X\) may separate the space in bad ways.\N\NThe heuristic idea is that quasiconvexity and good avoidance proprerties together yield Theorem 1.2. This is precisely what the assumptions of Ahlfors regularity and Poincaré inequality are capturing, as Sections 2--4 of the paper show. It is well-known that the latter implies quasiconvexity, while the Ahlfors regularity and restriction on the exponent \(p\) imply avoidance properties. See Corollary 3.4 for a nice quantification of this. Even armed with this intuition, the task of extending \(f\) is highly nontrivial. As we add curves to the gaps of \(f\), the bi-Lipschitz constants deteriorate. How then to ensure, that the constants do not blow up, as we need to glue infinitely many such curves? Further, Corollary 3.4 does not quite give a curve that goes all the way -- just one that almost does the job.\N\NThese are the main technical issues that are solved in Sections 5 and 6 of the paper. The proof proceeds by a multistep extension involving Whitney scales of the complement \(\mathbb{R}\setminus A\). First, a rough expansion is done for the end points of the Whitney intervals in Proposition 5.2. The key technical idea is to group the points into a bounded number of well-separated groups (Lemma 5.3) and then extend the map to each group simultaneously. Each group can be extended with a fixed increase in the bi-Lipschitz constants, and the extension is constructed by a finite number of such steps. The same idea is used to fill in the middle thirds of each Whitney interval, see Proposition 5.4 and Lemma 5.6. The proof of Theorem 1.2. is then obtained by connecting neighboring pieces and minor adjustments. This involves Lemma 4.2.\N\NThe proof is well presented and very carefully written. It illustrates clearly the main technical difficulties and is divided into steps in a transparent way. Despite the several layers of the procedure, the key technical insight seem to be the following:\N\begin{itemize}\N\item[1.] Corollary 3.4, which gives the relevance of the Poincaré inequality.\N\item[2.] Lemmas 4.1--4.2 which give bi-Lipschitz curves from bounded length curves.\N\item[3.] The proofs of Propositions 5.2 and 5.4, which guarantee that inifinitely many partial extensions can be performed simultaneously as long as their domains/images are well separated. And Lemmas 5.3 and 5.6, which guarantee that the extension steps can be grouped into boundedly many groups that have this separation condition.\N\end{itemize}\NIn many ways it is this last step, which prevents the bi-Lipschitz constant from growing beyond limit.