## GeoMeG: Description

The general goal of the project is to study Lie groups equipped with arbitrary distance functions and to develop an adapted geometric measure theory in the non-Riemannian settings.

The main focus is subRiemannian geometry, together with implications to control systems and nilpotent groups. SubRiemannian spaces, and especially Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, subelliptic PDE's and geometric group theory.

The members of the team master methods coming from geometric measure theory, geometric analysis, and group theory; in particular, analysis on metric spaces and Lie group theory.

This project has received funding from the European Research Council (ERC) under the European Union’s ERC Starting Grant (Grant agreement 713998 GeoMeG)

| Grant Agreement | University of Jyvskäylä Press Release |

## People

### Principal Investigator

Enrico Le Donne

| Personal website |

### Post-Docs

Daniela Di Donato
Sebastiano Don
Laurent Dufloux
Thibaut Dumont
Rami Luisto
Gabriel Pallier
Francesca Tripaldi

### Ph.D. Students

Gioacchino Antonelli
Eero Hakavuori
Ville Kivioja
Terhi Moisala
Alessandro Pilastro

## Jyväskylä Geometry Seminar

The Jyväskylä Geometry seminar is on Mondays afternoons
in the Department of Mathematics and Statistics.
The meeting is usually from 14:15 to 16:00 in room MaD 380.

No seminars are preview for the next weeks.

See all current Seminars | Analysis Seminar |

Past Seminars: 2021 | 2020 | 2019 | 2018 | 2017 | (older) |

## Future Events

17th-14th April 2030 | world wide web
International sub-Riemannian seminar
| Website |

No events are planned for the next months.

 Past Events February/2018 Sub-Riemannian Geometry and Beyond February/2019 Sub-Riemannian Geometry and Beyond, II August/2019 Jyväskylä Summer School - Courses in Geometry October/2019 Pansu's Fest, for the 60th birthday of Pierre Pansu May/2020 Sub-Riemannian Geometry and Beyond, III --- POSTPONED world wide web

## Recent Papers

We warmly thank the CVGMT Team for the support and
for hosting the following papers at CVGMT server.

 Nilpotent groups and biLipschitz embeddings into $L^1$ (2022) S. Eriksson-Bique, C. Gartland, E. Le Donne, L. Naples, S. Nicolussi-Golo
 On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion (2022) G. Antonelli, A. Merlo   (Journal of Functional Analysis)    In this paper we continue the study of the notion of $\mathscr{P}$-rectifiability in Carnot groups. We say that a Radon measure is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. In this paper we prove a Marstrand--Mattila rectifiability criterion in arbitrary Carnot groups for $\mathscr{P}$-rectifiable measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if a priori the tangent planes at a point might not be the same at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup of a Carnot group has a normal complement, our criterion applies in the particular case in which the tangents are one-dimensional horizontal subgroups. Hence, as an immediate consequence of our Marstrand--Mattila rectifiability criterion and a result of Chousionis--Magnani--Tyson, we obtain the one-dimensional Preiss's theorem in the first Heisenberg group $\mathbb H^1$. More precisely, we show that a Radon measure $\varphi$ on $\mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure $\mathcal{H}^1$, and it is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps from $A\subset \mathbb R$ to $\mathbb H^1$.
 Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity (2022) S. Nardulli, R. Resende    In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of $(C_0,\alpha_0,r_0)$-almost area minimizing currents of arbitrary dimension m and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current $T$ is an open dense set in $\Gamma$.
 Local minimizers and Gamma-convergence for nonlocal perimeters in Carnot Groups (2021) A. Carbotti, S. Don, D. Pallara, A. Pinamonti   (ESAIM cocv)    We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.
 Direct limits of infinite-dimensional Carnot groups (2021) T. Moisala, E. Pasqualetto   (Mathematica Scandinavica)    We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.
 Lecture notes on sub-Riemannian geometry from the Lie group viewpoint (2021) E. Le Donne    These are lecture notes from various courses on sub-Riemannian geometry. The viewpoint that is emphasised is the one from the Lie group theory and metric geometry.

| See the complete list of papers |

## Open Positions

Please, contact the PI if you are interested to do a Ph.D. or a postdoc
in the framework of GeoMeG ERC project.

New positions will be open in the next years.

Prof. Enrico Le Donne

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