76 research outputs found
Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics
We prove a classification theorem for conformal maps with respect to the
control distance generated by a system of diagonal vector fields.
It turns out that all such maps can be obtained as compositions of suitable
dilations, inversions and isometries. We also classify all umbilical surfaces
of the underlying metric.Comment: Revised version, to appear on Israel Journal of Mathematics. New
title and added section 4 on umbilical surface
Pseudohermitian invariants and classification of CR mappings in generalized ellipsoids
We discuss the problem of classifying all local CR diffeomorphisms of a
strictly pseudoconvex surface. Our method exploits the Tanaka--Webster
pseudohermitian invariants, their transformation formulae, and the Chern--Moser
invariants. Our main application concerns a class of generalized ellipsoids
where we classify all local CR mappings.Comment: Accepted version, to appear on J. Math. Soc. Japa
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
Stability of isometric maps in the Heisenberg group
In this paper we prove some approximation results for biLipschitz maps in the
Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz
constant close to one can be pointwise approximated, quantitatively on any
fixed ball, by an isometry. This leds to an approximation in BMO norm for the
map's Pansu derivative. We also prove that a global quasigeodesic can be
approximated by a geodesic in any fixed segment
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 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
Theory of activated-rate processes under shear with application to shear-induced aggregation of colloids
Using a novel approximation scheme within the convective diffusion (two body
Smoluchowski) equation framework, we unveil the shear-driven aggregation
mechanism at the origin of structure-formation in sheared colloidal systems.
The theory, verified against numerics and experiments, explains the induction
time followed by explosive (irreversible) rise of viscosity observed in
charge-stabilized colloidal and protein systems under steady shear. The
Arrhenius-type equation with shear derived here, extending Kramers theory in
the presence of shear, is the first analytical result clearly showing the
important role of shear-drive in activated-rate processes as they are
encountered in soft condensed matter
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
Isoperimetric inequality in the Grushin plane
We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric set
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