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On the cut locus of free, step two Carnot groups

Luca Rizzi 1, 2 Ulysse Serres 3
1 GECO - Geometric Control Design
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : In this note, we study the cut locus of the free, step two Carnot groups G k with k generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Mya02, Mya06] and [MM16a], by exhibiting sets of cut points C k ⊂ G k which, for k ≥ 4, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension Θ(k 2) and semi-algebraic sets of codimension Θ(k), the sets C k are semi-algebraic and have codimension 2, yielding the best possible lower bound valid for all k on the size of the cut locus of G k. Furthermore, we study the relation of the cut locus with the so-called abnormal set. In the low dimensional cases, it is known that Abn0(G k) = Cut0(G k) \ Cut0(G k), k = 2, 3. For each k ≥ 4, instead, we show that the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of C k. The question whether C k coincides with the cut locus for k ≥4 remains open.
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Luca Rizzi, Ulysse Serres. On the cut locus of free, step two Carnot groups. Proceedings of the American Mathematical Society, American Mathematical Society, 2017, 145, pp.5341-5357. ⟨10.1090/proc/13658⟩. ⟨hal-01377408v2⟩

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