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Metric-measure inequalities in sub-Riemannian manifolds

Periodic Reporting for period 1 - MesuR (Metric-measure inequalities in sub-Riemannian manifolds)

Reporting period: 2019-09-01 to 2021-08-31

MesuR has been devoted to deepen our knowledge of geometric and dynamical properties of a class of metric-measure spaces, called sub-Riemannian (sR) manifolds.

From a scientific point of view, MesuR allowed us to obtain new isoperimetric inequalities in a class of singular structures and to identify a purely sR behavior in a class of sR inequalities, called Hardy inequalities, and in the heat/quantum confinement of sR manifolds. We also treated heat kernel estimates in sR manifolds when a magnetic field influences the dynamic: this study was not previewed in the DoA, but it naturally arises as an open problem connected to MesuR's Objectives. Finally, I had the chance to open my research field to a new theme and collaborate on the applications of sR model of the visual cortex to human contrast perception.

From a wider perspective MesuR offered me the opportunity to to disseminate the results of the project to a wide (scientific and non-scientific) public, to deepen and enlarge my competences, produce high-level research and compete for positions in academia. This is supported by the fact that, during the first year of MesuR, I won a position as Researcher (RTD-A) at the University of Padova, that started in September 2020.
"MesuR focused on studying the following topics:

(I1). sR isoperimetric inequalities in a class of singular structures and their relations with the standing Pansu's conjecture about the shape of isoperimetric sets in the Heisenberg group. (I1) refers to Objectives (O1-O2). The obtained results are contained in a preprint in collaboration with R. Monti, A. Righini and M. Sigalotti which has been submitted for publication on a a top-level peer reviewed scientific journal.
In this work, we study the isoperimetric problem on the most studied 3D Carnot group, called the Heisenberg group, endowed with a non-regular sR structure (also called sub-Finsler structure). We obtain a precise relation between ""singular"" isoperimetric sets (see the attached picture) and already known geodesic properties. Although the singularity here has not to be intended in the volume measure, in a future work, we aim at adapting the novel use of adapted control theoretical techniques also to the case of singular (volume) measures.

(I2). Hardy inequalities in the Heisenberg group relative to the intrinsic distance. (I2) refers to Objective (O3). The obtained results are contained in a paper in collaboration with D. Prandi.
In this work, we show that Hardy inequalities in regular sR settings cannot have a Riemannian behavior when associated the sR distance. This work allowed us to specify a basic behavior of these metric-measure inequalities and it will allow us to better address the study of more general structures where singularities appear also in the measure, such as multi-dimensional Grushin structures.

(I3). Heat/quantum confinement in 3D, regular sR manifolds endowed with intrinsic measures. (I3) refers to Objective (O4).
The obtained results are contained in a paper in collaboration with R. Adami, U. Boscain, and D.Prandi which has been submitted for publication on a top-level peer reviewed scientific journal. In this work, we establish heat and quantum confinement in pointed 3D sR manifolds when the point is assumed to be regular. The results of the paper suggest that sR heat confinement behaves in a neatly different way with respect to the Riemannian one and sheds some light on the attended results in the case of the more general singular sub-Laplacians.

(I4). Heat kernel estimates in the Heisenberg group under the action of magnetic potentials. (I4) refers to Objectives (O4) and partly (O5). The obtained results are the subject of an ongoing collaboration with researchers working in France, Czech Republic and Italy. In the project we study how to suitably define magnetic potentials associated with the sR heat diffusion. We then study how the associated decay estimates are affected by the magnetic field and compare it to the Euclidean setting.

Guided by the Supervisor U.~Boscain, I also had the possibility to acquire new competences concerning the application of sR geometry to model the human vision, as mentioned in the Training Objective 3 - see Section 1.2.1 of DoA. As a consequence, I also worked on the following topic

(I5).Consistence of an existent sR model of human vision on modeling human contrast perception.
The results are contained in the remaining two papers cited in the Publications section

We committed to disseminate the results of MesuR to the widest scientific community and to different target audiences with the following strategies:

*I created a website to record the progress of MesuR and upload preprints, papers, slides.

*In collaboration with some coauthors we submitted the 5 papers mentioned above, three of which are published on peer reviewed journals.

*I participated to 4 conferences: in Oberwolfach - DE (October 19), Paris - FR (October 19 and November 19), Trento - IT (February 20). I also delivered 5 talks on the subjects of MesuR. Three more were previewed, but canceled due to the COVID-19 situation. I also participated to 3 scientific visits in Prague - CZE, Padova - IT and Grenoble - FR.

*I co-organized a conference in Paris in October 2019 that allowed different communities working on functional spectral analysis communicate between each other. I also co-organized the monthly seminar ``SGAS - Seminaire de G\'eom\'etrie et Analyse sous-Riemannienne'' to disseminate recent results in sR theory in the french community. Due to the COVID-19 situation, starting from April 2020 the SGAS seminar was held online with an international audience.
We planned to organize a workshop held in April 2020 in Paris on the modelization of human vision, but we need to reschedule it to a later date due to the COVID-19 situation.

*With the aim of promoting scientific studies and the possibility of carreer in research, in September 2019 I participated in the organization of a week of courses and discussions for the students beginning a Master Programme in Universit\'e Paris Sud, Orsay FR.

*With the aim of creating awareness of the role of science for the society, I participated to the project ""Penne amiche della scienza"" in which I was associated with a class of students at an Elementary school in Italy and discussed with them (by exchanging emails) about several themes concerning science and research."
The progress of MesuR beyond the state of the art concerns 1) sR isoperimetric inequalities and the 2) heat/quantum behavior of sR manifold.

1) Thanks to MesuR it is now possible to generalize Pansu's conjecture to a wider class of non-regular structures (called sub-Finsler ones). We mention that the problem is now of a great interest: in addition to MesuR, a research group in Granada recently took interest on the problem and obtained some results complementary to the ones of MesuR.

2) Thanks to MesuR we now have clear examples in which purely sub-Riemannian behaviors come into play: in the Heisenberg group the role of the sR distance in Hardy inequalities is not the same as in the Riemannian case, and 3D sR manifolds without a regular point are quantum confined, differently from the Riemannian case. This will allow for a further research considering singular sR manifolds to complete the full picture.

The project ended in August 2020 so that no more results are expected.