Periodic Reporting for period 2 - GeoSub (Geometric analysis of sub-Riemannian spaces through interpolation inequalities)
Reporting period: 2023-07-01 to 2024-12-31
Sub-Riemannian spaces are geometrical structures that model constrained systems and constitute a vast generalization of Riemannian geometry. They arise in control theory, harmonic and complex analysis, subelliptic PDEs, geometric measure theory, calculus of variations, optimal transport, and potential analysis. In the last 10 years, a surge of interest in the study of geometric and functional inequalities on sub-Riemannian spaces revealed unexpected behaviours and intriguing phenomena that failed to fit into the classical schemes inspired by Riemannian geometry. In this project, I aim to develop a framework of geometric and functional interpolation inequalities adapted to sub-Riemannian manifolds, and to use this theory to tackle old and new problems concerning the geometric analysis of these structures. The project focuses on the following interconnected topics:
(i) the development of a unifying theory of curvature bounds including sub-Riemannian structures,
(ii) the study of measure contraction properties of Carnot groups,
(iii) applications to isoperimetric-type problems,
(iv) applications to the regularity of the sub-Riemannian heat kernel at the cut locus.
The project adopts a unique approach combining methods from geometric control theory, optimal transport and comparison geometry that I developed in recent years, and which already allowed me and my collaborators to obtain important results in the field. The project aims to achieve an ambitious unification program, solve long-standing problems, and explore new research directions in sub-Riemannian geometry, with an impact in several neighboring areas, including geometric analysis on non-smooth spaces, analysis of hypoelliptic operators, geometric measure theory, spectral geometry. My long-term purpose is to build a leading research group in sub-Riemannian geometry, to significantly advance our understanding of geometry under non-holonomic constraints.
(i) the development of a unifying theory of curvature bounds including sub-Riemannian structures,
(ii) the study of measure contraction properties of Carnot groups,
(iii) applications to isoperimetric-type problems,
(iv) applications to the regularity of the sub-Riemannian heat kernel at the cut locus.
The project adopts a unique approach combining methods from geometric control theory, optimal transport and comparison geometry that I developed in recent years, and which already allowed me and my collaborators to obtain important results in the field. The project aims to achieve an ambitious unification program, solve long-standing problems, and explore new research directions in sub-Riemannian geometry, with an impact in several neighboring areas, including geometric analysis on non-smooth spaces, analysis of hypoelliptic operators, geometric measure theory, spectral geometry. My long-term purpose is to build a leading research group in sub-Riemannian geometry, to significantly advance our understanding of geometry under non-holonomic constraints.
A first research direction was the development of sub-Riemannian interpolation inequalities. The main result in this subject is the research monograph ``Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures’’, where we introduce the concept of gauge metric measure space. Furthermore, we achieved a complete proof of the ``no-CD theorem” for sub-Riemannian structures, which covers all cases remained open in the literature. The group also continued the development of the smooth sub-Riemannian comparison theory for H-type foliations and studied structural properties of the sub-Riemannian exponential map. Finally, we applied methods from sub-Riemannian comparison theory to contact geometry, developing a new method to detect overtwisted disks in contact sub-Riemannian structures.
A second direction of work was towards the study of the measure contraction properties (MCP) of sub-Riemannian structures in the large. We achieved a complete description of the MCP of the sub-Finsler Heisenberg group. Furthermore, an instance of the rigidity phenomenon was discovered, where minimality of the curvature exponent for sub-Finsler structures on the Heisenberg group implies that the structure is the standard sub-Riemannian one. The group also worked on the sub-Riemannian Weyl’s tube, obtaining (i) optimal regularity results for the distance from submanifolds; (ii) the asymptotic of the volume and perimeter of tubular neighborhood; (iii) for the case of curves in the Heisenberg groups, we proved that the volume of tubes around curves does not depend on the way the curve is isometrically embedded, but only on its Reeb angle. This last result is new even for the three-dimensional Heisenberg group.
Furthermore, we studied heat kernels in sub-Riemannian geometry, obtaining a fourth-order asymptotic expansion of the relative heat content of non-characteristic domains, and we studied the Weyl’s law for a large class of singular Riemannian manifolds including almost-Riemannian geometries.
An additional research direction, relevant for the main themes of the project, is on the so-called Sard conjecture. The main product of this work is the development of a new methodology to obtain Sard properties for maps from infinite dimensions. As an application, we proved a version of the sub–Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of analytic controls with large radius of convergence.
A second direction of work was towards the study of the measure contraction properties (MCP) of sub-Riemannian structures in the large. We achieved a complete description of the MCP of the sub-Finsler Heisenberg group. Furthermore, an instance of the rigidity phenomenon was discovered, where minimality of the curvature exponent for sub-Finsler structures on the Heisenberg group implies that the structure is the standard sub-Riemannian one. The group also worked on the sub-Riemannian Weyl’s tube, obtaining (i) optimal regularity results for the distance from submanifolds; (ii) the asymptotic of the volume and perimeter of tubular neighborhood; (iii) for the case of curves in the Heisenberg groups, we proved that the volume of tubes around curves does not depend on the way the curve is isometrically embedded, but only on its Reeb angle. This last result is new even for the three-dimensional Heisenberg group.
Furthermore, we studied heat kernels in sub-Riemannian geometry, obtaining a fourth-order asymptotic expansion of the relative heat content of non-characteristic domains, and we studied the Weyl’s law for a large class of singular Riemannian manifolds including almost-Riemannian geometries.
An additional research direction, relevant for the main themes of the project, is on the so-called Sard conjecture. The main product of this work is the development of a new methodology to obtain Sard properties for maps from infinite dimensions. As an application, we proved a version of the sub–Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of analytic controls with large radius of convergence.
All the results mentioned above constitute a progress beyond the state of the art for the corresponding research lines. We give an overview of the expected results until the end of the project:
- continue the study of gauge metric measure spaces, and the effect of abnormal geodesics in interpolation inequalities;
- study the MCP of spaces containing abnormal geodesics;
- work towards the understanding of isoperimetric-type inequalities in sub-Riemannian geometry;
- further develop the Sard-type results obtained so-far in the setting of Carnot groups.
- continue the study of gauge metric measure spaces, and the effect of abnormal geodesics in interpolation inequalities;
- study the MCP of spaces containing abnormal geodesics;
- work towards the understanding of isoperimetric-type inequalities in sub-Riemannian geometry;
- further develop the Sard-type results obtained so-far in the setting of Carnot groups.