3,997 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
Review: A Coherent and Comprehensive Model of the Evolution of the Outer Solar System
Since the discovery of the first extra-solar planets, we are confronted with
the puzzling diversity of planetary systems. Processes like planet radial
migration in gas-disks and planetary orbital instabilities, often invoked to
explain the exotic orbits of the extra-solar planets, at first sight do not
seem to have played a role in our system. In reality, though, there are several
aspects in the structure of our Solar System that cannot be explained in the
classic scenario of in-situ formation and smooth evolution of the giant
planets. This paper describes a new view of the evolution of the outer Solar
System that emerges from the so-called 'Nice model' and its recent extensions.
The story provided by this model describes a very "dynamical" Solar System,
with giant planets affected by both radial migrations and a temporary orbital
instability. Thus, the diversity between our system and those found so far
around other stars does not seem to be due to different processes that operated
here and elsewhere, but rather stems from the strong sensitivity of chaotic
evolutions to small differences in the initial and environmental conditions.Comment: in press in CR Physique de l'Academie des Science
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
Spin-Spin Coupling in the Solar System
The richness of dynamical behavior exhibited by the rotational states of
various solar system objects has driven significant advances in the theoretical
understanding of their evolutionary histories. An important factor that
determines whether a given object is prone to exhibiting non-trivial rotational
evolution is the extent to which such an object can maintain a permanent
aspheroidal shape, meaning that exotic behavior is far more common among the
small body populations of the solar system. Gravitationally bound binary
objects constitute a substantial fraction of asteroidal and TNO populations,
comprising systems of triaxial satellites that orbit permanently deformed
central bodies. In this work, we explore the rotational evolution of such
systems with specific emphasis on quadrupole-quadrupole interactions, and show
that for closely orbiting, highly deformed objects, both prograde and
retrograde spin-spin resonances naturally arise. Subsequently, we derive
capture probabilities for leading order commensurabilities and apply our
results to the illustrative examples of (87) Sylvia and (216) Kleopatra
asteroid systems. Cumulatively, our results suggest that spin-spin coupling may
be consequential for highly elongated, tightly orbiting binary objects.Comment: 9 pages, 4 figures, accepted to Ap
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
Dynamical Evolution Induced by Planet Nine
The observational census of trans-Neptunian objects with semi-major axes
greater than ~250 AU exhibits unexpected orbital structure that is most readily
attributed to gravitational perturbations induced by a yet-undetected, massive
planet. Although the capacity of this planet to (i) reproduce the observed
clustering of distant orbits in physical space, (ii) facilitate dynamical
detachment of their perihelia from Neptune, and (iii) excite a population of
long-period centaurs to extreme inclinations is well established through
numerical experiments, a coherent theoretical description of the dynamical
mechanisms responsible for these effects remains elusive. In this work, we
characterize the dynamical processes at play, from semi-analytic grounds. We
begin by considering a purely secular model of orbital evolution induced by
Planet Nine, and show that it is at odds with the ensuing stability of distant
objects. Instead, the long-term survival of the clustered population of
long-period KBOs is enabled by a web of mean-motion resonances driven by Planet
Nine. Then, by taking a compact-form approach to perturbation theory, we show
that it is the secular dynamics embedded within these resonances that regulates
the orbital confinement and perihelion detachment of distant Kuiper belt
objects. Finally, we demonstrate that the onset of large-amplitude oscillations
of orbital inclinations is accomplished through capture of low-inclination
objects into a high-order secular resonance and identify the specific harmonic
that drives the evolution. In light of the developed qualitative understanding
of the governing dynamics, we offer an updated interpretation of the current
observational dataset within the broader theoretical framework of the Planet
Nine hypothesis.Comment: 22 pages, 13 figures, accepted for publication in the Astronomical
Journa
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
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