Blowups and blowdowns of geodesics in Carnot groups (Q6158170)

Let \(G\) be a Carnot group of step \(s\) equipped with a left-invariant sub-Finsler metric. Given a Lipschitz curve \(\gamma\) in \(G\), one can construct its blow-up in a natural way: Supposing for instance that \(\gamma(0) = 1_G\) is the identity element, we let \(\gamma_h(t) = \delta_{1/h}(\gamma(ht))\), so that as \(h \to 0\), we are dilating the image of the curve while slowing down time to keep the speed approximately constant. Any limit of a sequence \(\gamma_{h_n}\) as \(h_n \to 0\), in the topology of uniform convergence on compact subsets of time, is called a \textit{tangent} of \(\gamma\). Every Lipschitz curve has at least one tangent (by Arzelá-Ascoli theorem), but in general it is not unique. We also define asymptotic cones (blow-downs) of \(\gamma\) in a similar way by letting \(h_n \to \infty\) instead. The first main result of this paper concerns projections of blow-ups. Let \(V_s\) be the top layer of the stratification of the Lie algebra of \(G\), which generates the subgroup \(\exp(V_s) \subset G\). Write \(\pi_{s-1}\) for the quotient map from \(G\) to \(G/\exp(V_s)\). Theorem 1.3 states that if \(\gamma\) is a geodesic and \(\sigma\) is a tangent to \(\gamma\) (which implies that \(\sigma\) is also a geodesic), then \(\pi_{s-1} \circ \sigma\) is a geodesic in \(G / \exp(V_s)\) with respect to the canonical sub-Finsler metric induced from \(G\). Keep in mind that the projection of a geodesic is in general not a geodesic. Iterating this \(s-1\) times, one has the corollary that any \((s-1)\)-fold iterated tangent \(\sigma\) of \(\gamma\) projects to a geodesic in the abelianization \(G / [G,G]\). When our sub-Finsler metric is actually sub-Riemannian, this means that \(\sigma\) is a line (Corollary 1.4). As an extension of this theorem, by considering nilpotent approximations, the authors show that in a general sub-Riemannian manifold of step \(s\) (not necessarily a Carnot group), any \((s-1)\)-fold iterated tangent of a geodesic is a line (Theorem 1.1). The last main result, back in the setting of a sub-Finsler Carnot group \(G\), deals with the projection \(\pi \circ \gamma\) into the abelianization \(G / [G,G]\) of an infinite geodesic \(\gamma : \mathbb{R} \to G\) (which is required to be length-minimizing along its entire length). It is shown that if \(\pi \circ \gamma\) is not itself a geodesic, then it still must lie within some distance \(R\) of some hyperplane \(W\) (Theorem 1.5). In the special case where the sub-Finsler metric is actually sub-Riemannian, this implies that all asymptotic cones of \(\gamma\) lie in a proper Carnot subgroup of \(G\) (Corollary 1.6).