104 research outputs found
An area formula in metric spaces
We present an area formula for continuous mappings between metric spaces,
under minimal regularity assumptions. In particular, we do not require any
notion of differentiability. This is a consequence of a measure theoretic
notion of Jacobian, defined as the density of a suitable "pull-back measure"
Contact equations, Lipschitz extensions and isoperimetric inequalities
We characterize locally Lipschitz mappings and existence of Lipschitz
extensions through a first order nonlinear system of PDEs. We extend this study
to graded group-valued Lipschitz mappings defined on compact Riemannian
manifolds. Through a simple application, we emphasize the connection between
these PDEs and the Rumin complex. We introduce a class of 2-step groups,
satisfying some abstract geometric conditions and we show that Lipschitz
mappings taking values in these groups and defined on subsets of the plane
admit Lipschitz extensions. We present several examples of these groups, called
Allcock groups, observing that their horizontal distribution may have any
codimesion. Finally, we show how these Lipschitz extensions theorems lead us to
quadratic isoperimetric inequalities in all Allcock groups.Comment: This version has additional references and a revisited introductio
Towards a theory of area in homogeneous groups
A general approach to compute the spherical measure of submanifolds in
homogeneous groups is provided. We focus our attention on the homogeneous
tangent space, that is a suitable weighted algebraic expansion of the
submanifold. This space plays a central role for the existence of blow-ups.
Main applications are area-type formulae for new classes of smooth
submanifolds. We also study various classes of distances, showing how their
symmetries lead to simpler area and coarea formulas. Finally, we establish the
equality between spherical measure and Hausdorff measure on all horizontal
submanifolds.Comment: 60 page
Nonexistence of horizontal Sobolev surfaces in the Heisenberg group
Involutivity is a well known necessary condition for integrability of smooth
tangent distributions. We show that this condition is still necessary for
integrability with Sobolev surfaces. We specialize our study to the left
invariant horizontal distribution of the first Heisenberg group \H^1. Here we
answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger,
J.T.Tyson
Blow-up of regular submanifolds in Heisenberg groups and applications
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group,
where intrinsic dilations are used. Main consequence of this result is an
explicit formula for the density of (p+1)-dimensional spherical Hausdorff
measure restricted to a p-dimensional submanifold with respect to the
Riemannian surface measure. We explicitly compute this formula in some simple
examples and we present a lower semicontinuity result for the spherical
Hausdorff measure with respect to the weak convergence of currents. Another
application is the proof of an intrinsic coarea formula for vector-valued
mappings on the Heisenberg group
A new differentiation, shape of the unit ball and perimeter measure
We present a new blow-up method that allows for establishing the first
general formula to compute the perimeter measure with respect to the spherical
Hausdorff measure in noncommutative nilpotent groups. This result leads us to
an unexpected relationship between the area formula with respect to a distance
and the profile of its corresponding unit ball.Comment: 17 page
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