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Francesca Tripaldi

Publications and source records attributed to Francesca Tripaldi.

At least 19 recordsLinked to original sources
arXiv2026

Spectral complexes and currents on the Engel group

We study the spectral complexes arising from the de Rham complex on the Engel group $G$ viewed as a truncated multicomplex. We describe the two resulting complexes, analyse their compactly supported counterparts, and define the corresponding spectral currents by duality. We prove that the smooth spectral complexes embed naturally into the current complexes. Finally, we show that the Pansu pullback of a $W_{\mathrm{loc}}^{1,q}$ map $\varphi\colon G\to G$ with $q>7$ induces a morphism into the corresponding complexes of spectral currents.

math.DG↗
arXiv2026

Higher degree obstructions to quasiconformal equivalence

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

math.MG↗
arXiv2026

Stokes' theorem on positively graded groups

This paper studies the validity of Stokes' theorem for differential subcomplexes naturally adapted to the noncommutative geometry of positively graded Lie groups, with particular emphasis on Carnot groups. We introduce geometric conditions under which Stokes-type formulae hold for the Rumin complex and for a new family of spectral complexes associated with the homogeneous weight filtration of the de Rham complex. In particular, the spectral complexes allow us to recover the validity of Stokes' theorem on locally smooth intrinsic graphs. This is achieved by showing that the corresponding Stokes' formulae are governed entirely by the degree of the underlying submanifolds. Our approach also reveals that both the Rumin complex and the spectral complexes can be interpreted directly in terms of the classical de Rham complex through the Leibniz rule and integration over suitable classes of submanifolds, namely R-manifolds and spectral manifolds, respectively. Finally, motivated by this interaction between homogeneous weights and degrees of submanifolds, we propose a notion of current naturally adapted to these subcomplexes.

math.DG↗
arXiv2026

Pansu pullback and spectral complexes

In this paper, we prove the commutativity between the Pansu pullback of a smooth contact map between Carnot groups and the differentials appearing in the spectral complexes. As a direct application, we also present a way of "lifting" a Pansu derivative (viewed as a Lie algebra homomorphism) from Carnot groups to their central extensions.

math.DG↗
arXiv2026

Spectral complexes from truncated multicomplexes

This paper introduces a new construction of subcomplexes associated with a truncated multicomplex. Inspired by the machinery of spectral sequences, this construction yields a collection of interrelated subcomplexes whose differentials coincide with the spectral sequence differentials. These complexes refine the Rumin complex and retain the cohomology of the underlying multicomplex, providing a new tool for the study of subRiemannian geometry, particularly on Carnot groups.

math.AT↗
arXiv2024

Surgery and positive Bakry-\'Emery Ricci curvature

We consider the problem of preserving weighted Riemannian metrics of positive Bakry-\'Emery Ricci curvature along surgery. We establish two theorems of this type: One for connected sums, and one for surgeries along higher-dimensional spheres. In contrast to known surgery results for positive Ricci curvature, these results are local, i.e. we only impose assumptions on the weighted metric locally around the sphere along which the surgery is performed. As application we then show that all closed, simply-connected spin 5-manifolds admit a weighted Riemannian metric of positive Bakry-\'Emery Ricci curvature. By a result of Lott, this also provides new examples of manifolds with a Riemannian metric of positive Ricci curvature.

math.DG↗
arXiv2024

Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates

On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered by Rumin, who introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential $d_c$ is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.

math.AP↗
arXiv2023

Subcomplexes on filtered Riemannian manifolds

In this paper, we present a general construction to extract subcomplexes from two distinct complexes on filtered Riemannian manifolds. The first subcomplex computes the de Rham cohomology of the underlying manifold. On regular subRiemannian manifold equipped with a compatible Riemannian metric, it aligns locally with the so-called Rumin complex. The second complex instead generalises the Chevalley-Eilenberg complex computing Lie algebra cohomology of a nilpotent Lie group. Our approach offers key insights on the role of the Riemannian metric when extracting subcomplexes, opening up potential new applications in more general geometric settings, such as singular subRiemannian manifolds.

math.DG↗
arXiv2023

Filtered complexes and cohomologically equivalent subcomplexes

Inspired by Rumin's work on a subcomplex in sub-Riemannian manifolds which is cohomologically equivalent to the de Rham complex, we present a more general construction that produces subcomplexes from any filtered cochain complex of finite depth and still computes the cohomology of the original filtered complex. A priori these subcomplexes depend not only on the filtration itself, but also on the choice of additional structures. However, we show that the construction only depends on the given filtration up to isomorphism. Finally, we show how such subcomplexes relate to spectral sequences, a cohomological machinery that arises naturally when considering a filtered complex.

math.DG↗
arXiv2022

Sharp log-Sobolev inequalities in ${\sf CD}(0,N)$ spaces with applications

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

math.AP↗
arXiv2022

Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry

In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

math.DG↗
arXiv2020

Gradings for nilpotent Lie algebras

We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing $\ell^{q,p}$ cohomology.

math.GR↗
arXiv2020

The Rumin complex on nilpotent Lie groups

In this paper an alternative definition of the Rumin complex $(E_0^\bullet,d_c)$ is presented, one that relies on a different concept of weights of forms. In this way, the Rumin complex can be constructed on any nilpotent Lie group equipped with a Carnot-Carathéodory metric. Moreover, this construction allows for the direct application of previous non-vanishing results of $\ell^{q,p}$ cohomology to all nilpotent Lie groups that admit a positive grading.

math.DG↗
arXiv2020

A Cornucopia of Carnot groups in Low Dimensions

Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invariant 1-forms, a basis of right-invariant vector fields, and some other properties. We exhibit all stratified groups in dimension up to 7 and also study some free-nilpotent groups in dimension up to 14.

math.DG↗
arXiv2019

On the topology of surfaces with the generalised simple lift property

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised simple lift property.

math.GT↗
arXiv2019

Averages and the $\ell^{q,1}$-cohomology of Heisenberg groups

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.

math.DG↗