379 research outputs found
On the Whitney distortion extension problem for and and its applications to interpolation and alignment of data in
Let , open. In this paper we provide a sharp
solution to the following Whitney distortion extension problems: (a) Let
be a map. If is compact (with some
geometry) and the restriction of to is an almost isometry with small
distortion, how to decide when there exists a one-to-one and
onto almost isometry with small distortion
which agrees with in a neighborhood of and a Euclidean motion
away from . (b) Let
be map. If is compact (with some geometry) and the
restriction of to is an almost isometry with small distortion, how
to decide when there exists a one-to-one and onto
almost isometry with small distortion which
agrees with in a neighborhood of and a Euclidean motion away from . Our results complement those of [14,15,20]
where there, is a finite set. In this case, the problem above is also a
problem of interpolation and alignment of data in .Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451,
arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney
type extensions of distortions from certain compact sets to distorted diffeomorphisms on $\Bbb R^n
A Bounded mean oscillation (BMO) theorem for small distorted diffeomorphisms from to and PDE
This announcement considers the following problem. We produce a bounded mean
oscillation theorem for small distorted diffeomorphisms from to
. A revision of this announcement is in the memoir preprint:
arXiv:2103.09748, [1], submitted for consideration for publication.Comment: This paper appears as arXiv:1610.08138 which was submitted as a new
work by accident. Thus withdrawal is appropriat
On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in -Part 1
In this paper, we study the following problem: Let and let
be finite satisfying certain conditions. Suppose that we
are given a map with a small distortion on .
How can one decide whether extends to a smooth small distortion
which agrees with on . We also ask
how to decide if in addition can be approximated well by certain rigid
and non-rigid motions from . Since is a finite
set, this question is basic to interpolation and alignment of data in .Comment: This is part one of four papers with Charles Fefferman
(arXiv:1411.2468, arXiv:1505.06950, arXiv:1610.08138): Revised 1/13/201
Atoms and Quantum Dots With a Large Number of Electrons: the Ground State Energy
We compute the ground state energy of atoms and quantum dots with a large
number N of electrons. Both systems are described by a non-relativistic
Hamiltonian of electrons in a d-dimensional space. The electrons interact via
the Coulomb potential. In the case of atoms (d=3), the electrons are attracted
by the nucleus, via the Coulomb potential. In the case of quantum dots (d=2),
the electrons are confined by an external potential, whose shape can be varied.
We show that the dominant terms of the ground state energy are those given by a
semiclassical Hartree-exchange energy, whose N to infinity limit corresponds to
Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates
oscillations in the ground state energy as a function of N. These oscillations
reflect the dynamics of a classical particle moving in the presence of the
Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in
general in the case of dots, the motion contains a chaotic component. We
compute the correlation effects. They appear at the order N ln N for atoms, in
agreement with available data. For dots, they appear at the order N.Comment: 30 pages, 1 figur
Gravity, Two Times, Tractors, Weyl Invariance and Six Dimensional Quantum Mechanics
Fefferman and Graham showed some time ago that four dimensional conformal
geometries could be analyzed in terms of six dimensional, ambient, Riemannian
geometries admitting a closed homothety. Recently it was shown how conformal
geometry provides a description of physics manifestly invariant under local
choices of unit systems. Strikingly, Einstein's equations are then equivalent
to the existence of a parallel scale tractor (a six component vector subject to
a certain first order covariant constancy condition at every point in four
dimensional spacetime). These results suggest a six dimensional description of
four dimensional physics, a viewpoint promulgated by the two times physics
program of Bars. The Fefferman--Graham construction relies on a triplet of
operators corresponding, respectively to a curved six dimensional light cone,
the dilation generator and the Laplacian. These form an sp(2) algebra which
Bars employs as a first class algebra of constraints in a six-dimensional gauge
theory. In this article four dimensional gravity is recast in terms of six
dimensional quantum mechanics by melding the two times and tractor approaches.
This "parent" formulation of gravity is built from an infinite set of six
dimensional fields. Successively integrating out these fields yields various
novel descriptions of gravity including a new four dimensional one built from a
scalar doublet, a tractor vector multiplet and a conformal class of metrics.Comment: 27 pages, LaTe
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
Mass, Angular Momentum and Thermodynamics in Four-Dimensional Kerr-AdS Black Holes
In this paper, the connection between the Lorentz-covariant counterterms that
regularize the four-dimensional AdS gravity action and topological invariants
is explored. It is shown that demanding the spacetime to have a negative
constant curvature in the asymptotic region permits the explicit construction
of such series of boundary terms. The orthonormal frame is adapted to
appropriately describe the boundary geometry and, as a result, the boundary
term can be expressed as a functional of the boundary metric, extrinsic
curvature and intrinsic curvature. This choice also allows to write down the
background-independent Noether charges associated to asymptotic symmetries in
standard tensorial formalism. The absence of the Gibbons-Hawking term is a
consequence of an action principle based on a boundary condition different than
Dirichlet on the metric. This argument makes plausible the idea of regarding
this approach as an alternative regularization scheme for AdS gravity in all
even dimensions, different than the standard counterterms prescription. As an
illustration of the finiteness of the charges and the Euclidean action in this
framework, the conserved quantities and black hole entropy for four-dimensional
Kerr-AdS are computed.Comment: 15 pages,no figures,few references added,JHEP forma
Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces
In this paper we present a survey of the joint program with Fabrice Baudoin
originated with the paper \cite{BG1}, and continued with the works \cite{BG2},
\cite{BBG}, \cite{BG3} and \cite{BBGM}, joint with Baudoin, Michel Bonnefont
and Isidro Munive.Comment: arXiv admin note: substantial text overlap with arXiv:1101.359
A One-Dimensional Model for Many-Electron Atoms in Extremely Strong Magnetic Fields: Maximum Negative Ionization
We consider a one-dimensional model for many-electron atoms in strong
magnetic fields in which the Coulomb potential and interactions are replaced by
one-dimensional regularizations associated with the lowest Landau level. For
this model we show that the maximum number of electrons is bounded above by
2Z+1 + c sqrt{B}.
We follow Lieb's strategy in which convexity plays a critical role. For the
case of two electrons and fractional nuclear charge, we also discuss the
critical value at which the nuclear charge becomes too weak to bind two
electrons.Comment: 23 pages, 5 figures. J. Phys. A: Math and General (in press) 199
The CFT dual of AdS gravity with torsion
We consider the Mielke-Baekler model of three-dimensional AdS gravity with
torsion, which has gravitational and translational Chern-Simons terms in
addition to the usual Einstein-Hilbert action with cosmological constant. It is
shown that the topological nature of the model leads to a finite
Fefferman-Graham expansion. We derive the holographic stress tensor and the
associated Ward identities and show that, due to the asymmetry of the left- and
right-moving central charges, a Lorentz anomaly appears in the dual conformal
field theory. Both the consistent and the covariant Weyl and Lorentz anomaly
are determined, and the Wess-Zumino consistency conditions for the former are
verified. Moreover we consider the most general solution with flat boundary
geometry, which describes left-and right-moving gravitational waves on AdS_3
with torsion, and shew that in this case the holographic energy-momentum tensor
is given by the wave profiles. The anomalous transformation laws of the wave
profiles under diffeomorphisms preserving the asymptotic form of the bulk
solution yield the central charges of the dual CFT and confirm the results that
appeared earlier on in the literature. We finally comment on some points
concerning the microstate counting for the Riemann-Cartan black hole.Comment: 17 pages, uses JHEP3.cls. References added, minor errors correcte
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