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

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker "horizontal" sense.Comment: Revised version. To appear on J. Math. Anal- App

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

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

A Hadamard-type open map theorem for submersions and applications to completeness results in Control Theory

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical completeness result by Palais.Comment: 12 pages. Revised version. Minor changes. To appear on Annali di Matematic

Nonsmooth Hormander vector fields and their control balls

We prove a ball-box theorem for nonsmooth Hormander vector fields of step s.Comment: Final version. Trans. Amer. Math. Soc. (2012 to appear

Anisotropic estimates of subelliptic type

We discuss some estimates of subelliptic type related with vector fields satisfying the Hormander condition. Our approach makes use of a class of approximate exponentials studied in our previous papers. Such kind of estimates arises naturally in the study of regularity theory of weak solutions of degenerate elliptic equations

Multiexponential maps in Carnot groups with applications to convexity and differentiability

We analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems. First, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove the Pansu differentiability of the subRiemannian distance from a fixed point.Comment: Revised version. Published on Annali di Matematica Pura ed Applicat

Cosmological Measurements with General Relativistic Galaxy Correlations

We investigate the cosmological dependence and the constraining power of large-scale galaxy correlations, including all redshift-distortions, wide-angle, lensing and gravitational potential effects on linear scales. We analyze the cosmological information present in the lensing convergence and in the gravitational potential terms describing the so-called "relativistic effects," and we find that, while smaller than the information contained in intrinsic galaxy clustering, it is not negligible. We investigate how neglecting them does bias cosmological measurements performed by future spectroscopic and photometric large-scale surveys such as SKA and Euclid. We perform a Fisher analysis using the CLASS code, modified to include scale-dependent galaxy bias and redshift-dependent magnification and evolution bias. Our results show that neglecting relativistic terms introduces an error in the forecasted precision in measuring cosmological parameters of the order of a few tens of percent, in particular when measuring the matter content of the Universe and primordial non-Gaussianity parameters. Therefore, we argue that radial correlations and integrated relativistic terms need to be taken into account when forecasting the constraining power of future large-scale number counts of galaxy surveys.Comment: 18 pages, 10 figure