arXiv subjects
Publications and source records attributed to Pierre Pansu.
Pairs of diffeomorphic Riemannian manifolds are shown not to be quasiconformally equivalent, while no degree one invariant (extremal lengths, conformal parabolicity,...) can distinguish them. The obstruction arises from conformal cohomology in degree 2. Similar examples are constructed in the contact subRiemannian category
In this short note, we prove that all geodesically convex functions defined on a Riemannian manifold are continuous in the interior of their domain. This is a folklore result, but to the best of our knowledge, there is only one available proof, which is largely cited. However, it contains a significant gap, which we fill here. We also discuss extensions of this result beyond the Riemannian setting.
There are three approaches to currents tuned to the anisotropic geometry of Heisenberg groups: Ambrosio and Kirchheim's approach valid for general metric spaces; distributions dual to horizontal differential forms; distributions dual to Rumin's complex. It is shown that, in dimensions less than half the ambient dimension, these three theories coincide. On the other hand, they diverge beyond middle dimension: Ambrosio-Kirchheim currents vanish, Rumin currents correspond to a new class of Federer-Fleming currents called oblique currents.
We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more general results beyond the nilpotent case. In particular, we show that quasi-conformal homeomorphisms between geodesic Lie groups are quasi-isometries whenever the spaces have strict parabolic or hyperbolic conformal type. As a consequence, quasi-conformal homeomorphisms between geodesic Lie groups with infinite fundamental group are quasi-isometries. The statements for Lie groups are deduced from a more general study on metric measure spaces with uniformly locally bounded geometry.
We give alternative proofs of (unsharp) results of Gromov's on his H\''older equivalence problem: for which $\alpha$ does there exist a $C^\alpha$-homeomorphism of an open set of Euclidean space to an open set of a given Carnot group? We indicate a possible route to sharp bounds.
The aim of homotopy theory in topology is to simplify, after continuous deformation, continuous maps between topological spaces. What prevents this from happening are homotopy invariants. This raises quantitative questions: $\bullet$ Is the calculation of invariants possible (decidable)? If so, at what cost? $\bullet$ Is it possible to construct low-complexity representatives whose invariant values are prescribed? If so, at what cost? $\bullet$ How complex are the necessary deformations? The answers, often recent, are extremely varied. Moreover, many questions remain open, showing that topology has not said its last word, even in low dimensions.
In the Euclidean space it is known that a function $f\in L^2$ of a ball, with vanishing average,is the divergence of a vector field $F\in L^2$ with$$\| F\|\_{ L^2(B)} \le C \|f\|\_{L^2(B)}.$$In this Note we prove a similar result in any Carnot group $\mathbb{G}$ for a vanishing average $f\in L^p$, $1\le p < Q$, where $Q$ is the so-called homogeneous dimension of $\mathbb{G}$.
We show that every integer in the interval $[2p\chi(\Sigma), -2p\chi(\Sigma)]$ is achieved by the signature of a rank $2p$ flat symplectic bundle over a surface with boundary $\Sigma$. When $p=1$, one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.
It is shown that higher degree exact differential forms on compact Riemannian $n$-manifolds possess continuous primitives whose uniform norm is controlled by their $L^n$ norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms.
We prove that the family of normal currents in the sense of Rumin in a Carnot group is compact in the flat topology. This result is obtained through a dual compactness argument for Rumin forms, using the pseudo-differential calculus in groups developed by Folland, Christ-Geller-G lowacki-Polin and Rumin. As an application, imitating de Pauw-Moonens-Pfeffer, we describe the space of charges on a Carnot group.
This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems of Atiyah-Patodi-Singer, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on $\mathrm{U}(p)$, to discontinuous class functions, first on $\mathrm{U}(p,q)$, and then on other classical groups via embeddings into $\mathrm{U}(p,q)$. In this way, we present three different invariants -- signature, Toledo and rho invariant -- in a unifying way, which is a version of the classical signature formula of Atiyah-Patodi-Singer for manifolds with boundary.
In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.
In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups $\mathbb{H}^n$ for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that $d_c$ is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable $p$ and $q$, we mean that every exact differential form $ω$ in $L^p$ admits a primitive $ϕ$ in $L^q$ such that$\|ϕ\|_{L^{q}}\leq C\ \|ω\|_{L^{p}}$. The cases of the norm $L^p$, $p\ge 1$ and $q<\infty$ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where $q=\infty$: it is remarkable that, unlike in the scalar case, when the degree of the forms $ω$ is at least $2$, we can take $q=\infty$ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting $\mathbb{R}^N$ ($p=N$ and $q=\infty$) was proven by Bourgain and Brezis.
Averages are invariants defined on the $\ell^1$ cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the $\ell^1$ cohomology vanishes in these cases.
In this paper, we prove Poincaré and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.
In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in Heisenberg groups.
We study the simplicial {\ell} q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p, q) gap with respect to the weight gaps in the Lie algebra cohomology of G.
Roughly speaking, let us say that a map between metric spaces is large scale conformal if it maps packings by large balls to large quasi-balls with limited overlaps. This quasi-isometry invariant notion makes sense for finitely generated groups. Inspired by work by Benjamini and Schramm, we show that under such maps, some kind of dimension increases: exponent of volume growth for nilpotent groups, conformal dimension of the ideal boundary for hyperbolic groups. A purely metric space notion of {\ell} p-cohomology plays a key role.