On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

Montanari, Annamaria; Morbidelli, Daniele

doi:10.1016/j.jmaa.2016.07.032

Cited by 8 publications

(11 citation statements)

References 27 publications

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“…In fact, as proved in [ABCK97], Martinet spheres possess outward corners in correspondence of points reached by abnormal minimizing geodesics, and this implies the loss of semiconcavity. The same characterization holds for the Engel group (a step 3 and rank 2 Carnot structure on R 4 ), and for all free Carnot group of step 2, as proved in [MM16]. Finally, in [MM17], the authors proved the inclusion CutOpt(x) ⊆ SC + (x) for the free Carnot group of step 2 and rank 3.…”

Section: 21mentioning

confidence: 62%

Sub-Riemannian interpolation inequalities

Barilari

Rizzi

2018

Invent. math.

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We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As a byproduct, we characterize the cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex, answering to a question raised by Figalli and Rifford in [FR10].As an application, we deduce sharp and intrinsic Borell-Brascamp-Lieb and geodesic Brunn-Minkowski inequalities in the aforementioned setting. For the case of the Heisenberg group, we recover in an intrinsic way the results recently obtained by Balogh, Kristály and Sipos in [BKS18], and we extend them to the class of generalized H-type Carnot groups. Our results do not require the distribution to have constant rank, yielding for the particular case of the Grushin plane a sharp measure contraction property and a sharp Brunn-Minkowski inequality.

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Section: 21mentioning

confidence: 62%

Sub-Riemannian interpolation inequalities

Barilari

Rizzi

2018

Invent. math.

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“…However, it is known that the subRiemannian distance is smooth outside of the abnormal set and of the cut locus. In our model, it turns out that the abnormal set has the form Abn 0 = V × 0 ∧ 2 V (see Section 2.3) and it has been proved that the distance from the origin d is not differentiable at any abnormal point (see [MM16] for precise estimates). As a byproduct of our Theorem 3.2 it is easy to see that d is not differentiable at any point of Cut 0 (see Remark 3.7).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 93%

“…Concerning the three dimensional case, dim V = 3, it is well known that any bivector t ∈ ∧ 2 V can be written in the elementary form t = x ∧ y, for suitable x, y ∈ V. Although the choice of x and y is not unique, the subspace span{x, y} does not depend on such choice and it is called support of the bivector. See the discussion in [MM16], where the higher dimensional case is treated.…”

Section: Bivectorsmentioning

confidence: 99%

On the subRiemannian cut locus in a model of free two-step Carnot group

Montanari

Morbidelli

2017

Calc. Var.

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We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.Comment: Added Section 6. Final version, to appear on Calc. Va

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“…Remark 2.8. -We refer the reader to [21] for an other notion of horizontal semiconcavity of interest that has been investigated by Montanari and Morbidelli in the framework of Carnot groups.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%