## 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: 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.

 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    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.
 The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysis (2021) G. Antonelli, M. Fogagnolo, M. Pozzetta    In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.
 Ultralimits of pointed metric measure spaces (2021) E. Pasqualetto, T. Schultz    The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured Gromov-Hausdorff convergence. We also introduce a weaker variant of pointed measured Gromov-Hausdorff convergence, for which we can prove a version of Gromov's compactness theorem by using the ultralimit machinery. This compactness result shows that, a posteriori, our newly introduced notion of convergence is equivalent to the pointed measured Gromov one. Another byproduct of our ultralimit construction is the identification of direct and inverse limits in the category of pointed metric measure spaces.
 On rectifiable measures in Carnot groups: representation (2021) G. Antonelli, A. Merlo    This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of P-rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic C^1-rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic C^1-rectifiable.
 Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces (2020) G. Antonelli, E. Le Donne   (Nonlinear Analysis)    This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mathcal{H}^{12}$. As a consequence, we show that for every Carnot group of Hausdorff dimension 12, any Lipschitz map defined on a subset of it with values in $S$ has $\mathcal{H}^{12}$-null image. In particular, we deduce that this smooth hypersurface cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension $12$. As main consequence we have that a notion of rectifiability proposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioni and F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset $U$ of a homogeneous subgroup of Hausdorff dimension $12$ of a Carnot group, every bi-Lipschitz map $f:U\to S$ satisfies $\mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{\infty}$-hypersurfaces in $\mathbb H^n$ with $n\geq 2$ are countably $\mathbb{H}^{n-1}\times\mathbb R$-rectifiabile according to Pauls' definition, even with bi-Lipschitz maps.

| 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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