On the cut locus of free, step two Carnot groups
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- by Luca Rizzi and Ulysse Serres PDF
- Proc. Amer. Math. Soc. 145 (2017), 5341-5357
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 $k \geqslant 4$, 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 \[ \mathrm {Abn}_0(\mathbb {G}_k) = \overline {\mathrm {Cut}_0(\mathbb {G}_k)} \setminus \mathrm {Cut}_0(\mathbb {G}_k), \qquad k=2,3. \] For each $k \geqslant 4$, 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 $k\geqslant 4$ remains open.
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Additional Information
- Luca Rizzi
- Affiliation: Université Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France
- MR Author ID: 1037638
- Email: luca.rizzi@univ-grenoble-alpes.fr
- Ulysse Serres
- Affiliation: Université Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, 43 bd du 11 novembre 1918, F-69100 Villeurbanne, France
- Email: ulysse.serres@univ-lyon1.fr
- 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
- © Copyright The Authors, 2017, All Rights Reserved
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5341-5357
- MSC (2010): Primary 53C17, 49J15
- DOI: https://doi.org/10.1090/proc/13658
- MathSciNet review: 3717961