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On the cut locus of free, step two Carnot groups

Authors: Luca Rizzi and Ulysse Serres
Journal: Proc. Amer. Math. Soc. 145 (2017), 5341-5357
MSC (2010): Primary 53C17, 49J15
Published electronically: June 16, 2017
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Abstract: In this note, we study the cut locus of the free, step two Carnot groups $ \mathbb{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 works by Myasnichenko and Montanari and Morbidelli, by exhibiting sets of cut points $ C_k \subset \mathbb{G}_k$ which, for $ k \geq 4$, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension $ \Theta (k^2)$ and semi-algebraic sets of codimension $ \Theta (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 $ \mathbb{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

$\displaystyle \textup {Abn}_0(\mathbb{G}_k) = \overline {\textup {Cut}_0(\mathbb{G}_k)} \setminus \textup {Cut}_0(\mathbb{G}_k), \qquad k=2,3. $

For each $ k \geq 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\geq 4$ remains open.

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Luca Rizzi
Affiliation: Université Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France

Ulysse Serres
Affiliation: Université Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 bd du 11 novembre 1918, F-69100 Villeurbanne, France

Received by editor(s): October 10, 2016
Received by editor(s) in revised form: January 9, 2017
Published electronically: June 16, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright The Authors, 2017, All Rights Reserved

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