## Final Report Summary - CG-DICE (Dimension Phenomena and Curvature Equations in Carnot Groups)

The core of this project can be shortly (and roughly) centred on Geometric Measure Theory and curvature equations in non-Euclidean structures. It is worthwhile to state clearly that, when we mention non-Euclidean structures, we refer to metric structures that are non-Euclidean at any scale. Thus, the model we have in mind are not Riemannian manifolds, but better the so-called sub-Riemannian manifolds and fractals, or even fractals in sub-Riemannian spaces.

In the last few years, sub-Riemannian structures have been largely studied in several respects, such as differential geometry, geometric measure theory, subelliptic differential equations, complex variables, optimal control theory, mathematical models in neurosciences, non-holonomic mechanics, robotics.

Among all sub-Riemannian structures, a prominent position is taken by the so-called Carnot groups, which play versus sub Riemannian spaces the role played by Euclidean spaces (considered as tangent spaces) versus Riemannian manifolds.

The aim of this project was not centered on few specific points, but more in a wide-range analysis of the relationships between dimension gaps in the geometry of Carnot groups and other "pathological" phenomena of these structures. We particularly focused on phenomena related to different notions of curvatures, quasiconformal maps, differential forms and other problems in geometric measure theory.

Accordingly, the results obtained (about 110 publications) contain several progresses distributed in many areas.

In particular, we mention:

* We proved an estimate of the (Carnot-Carathéodory) Hausdorff dimension of the set generated by a self-similar iterated function systems in a Carnot groups.

* We described the action of projection operators in Carnot groups on Hausdorff dimensions and Hausdorff measure of subsets of the Heisenberg group, both notions being considered with respect to a sub-Riemannian metric.

* We introduced and studied a new notion of graph and, first of all, of Lipschitz graph within a Carnot group. This notion yielded an unifying point of view for regular submanifolds of Carnot groups, and has been shown to be enough ductile to be used both in rectifiability theory and in the study of partial differential equations.

* We studied minimal intrinsic graphs in Carnot groups associated with a decomposition of the group.

* We proved higher order isoperimetric inequalities on the boundary of a pseudoconvex domain (it is well known that Carnot groups appear also as models of pseudoconvex domains).

* We studied the evolution of surfaces in Carnot groups according to their intrinsic mean curvature, with applications to the modeling of the human vision.

* We proved existence results for the mean curvature equation in the hyperbolic plane, as well as symmetry results in terms of Levi mean curvature equation in real hypersurfaces of $C^{n+1}$.

* We proved the existence of non-smooth viscosity solutions to the Levi-curvature equations with smooth prescribed curvature.

* A new notion of Maxwell's equations in groups was introduced by some members of the project, relying on a notion of intrinsic complex of differential forms introduced by M. Rumin. Basically, this complex keeps into account the lack of commutativity of the generating vector fields in a Carnot group. It has been intriguing to discover that intrinsic Maxwell's equations yield a new class of "wave equations'', with new unexpected properties that reflect the group structure and its stratification. They are, in other words, another facet of the phenomena that appear in geometric measure theory as "dimension jumps''. We have been able to show that these equations can be viewed as limits of “real” Maxwell’s equations in the matter, in presence of strong anisotropies of its magnetic and electric properties. They are also deeply connected with the notion of quasiconformal maps in a group.

* We studied Lipschitz homotopy groups of the Heisenberg group with applications to the problem of density of Lipschitz mappings in the class of Sobolev mappings into the Heisenberg group.

* We constructed an approximately differentiable homeomorphism that preserves orientation, but whose Jacobian equals $-1$ a.e. Such a homeomorphism can be obtained as a limit of uniformly convergent diffeomorphisms.

* We introduced a new notion of convexity in Carnot groups. This subject is of crucial relevance both for its applications to geometric measure theory and to the general curvature problem. In particular, a correct notion of convexity is strictly connected with the properties of the Monge-Ampère equation (also studied by participants to the project) and to the applications to ABP inequality and regularity problems for nondivergence form pde’s in Carnot groups.

* Some of the participants to the project showed that in the Euclidean setting there is a deep connection between fractional Laplacian and generalized curvatures. In the last year, the same members of the project attacked the study of fraction Laplacians in Carnot groups, aiming to find an alternative approach to the problem of general curvatures in groups.

Some of the problems raised at the beginning of the project are still open:

* We are far from having developed a full theory of currents in Carnot groups and of related isoperimetric inequality, basically because of the lack of a theory of intrinsic Lipschitz maps in higher codimension, in spite of the contribution in codimension 1 obtained by the research group. A promising interesting recent remark is the relationship between intrinsic graphs and Rumin’s differential forms.

* In spite of some results on convex sets and non-divergence form equations in Carnot groups (ABP inequalities for non-divergence form equations are the pde's counterpart of the notion of curvature), a big issue is still open: "what is the Gauss curvature in a Carnot group?'' We are studying a possible approach to this crucial question that can be given by means of the so called Steiner formula.

During the duration of the project, the members of the different teams produced about 110 papers, that have been either published or accepted by high level international scientific journals (most of them belonging to the top 33% of the ISI category "Mathematics", according the Impact Factor). A full list of these papers can be found in the project website and in the attached list of publications and abstracts.

The researchers of the projects gave a large number of talks in conferences through the world, and organized several meetings and schools that are listed in the website (http://cg-dice.dm.unibo.it/).

In the last few years, sub-Riemannian structures have been largely studied in several respects, such as differential geometry, geometric measure theory, subelliptic differential equations, complex variables, optimal control theory, mathematical models in neurosciences, non-holonomic mechanics, robotics.

Among all sub-Riemannian structures, a prominent position is taken by the so-called Carnot groups, which play versus sub Riemannian spaces the role played by Euclidean spaces (considered as tangent spaces) versus Riemannian manifolds.

The aim of this project was not centered on few specific points, but more in a wide-range analysis of the relationships between dimension gaps in the geometry of Carnot groups and other "pathological" phenomena of these structures. We particularly focused on phenomena related to different notions of curvatures, quasiconformal maps, differential forms and other problems in geometric measure theory.

Accordingly, the results obtained (about 110 publications) contain several progresses distributed in many areas.

In particular, we mention:

* We proved an estimate of the (Carnot-Carathéodory) Hausdorff dimension of the set generated by a self-similar iterated function systems in a Carnot groups.

* We described the action of projection operators in Carnot groups on Hausdorff dimensions and Hausdorff measure of subsets of the Heisenberg group, both notions being considered with respect to a sub-Riemannian metric.

* We introduced and studied a new notion of graph and, first of all, of Lipschitz graph within a Carnot group. This notion yielded an unifying point of view for regular submanifolds of Carnot groups, and has been shown to be enough ductile to be used both in rectifiability theory and in the study of partial differential equations.

* We studied minimal intrinsic graphs in Carnot groups associated with a decomposition of the group.

* We proved higher order isoperimetric inequalities on the boundary of a pseudoconvex domain (it is well known that Carnot groups appear also as models of pseudoconvex domains).

* We studied the evolution of surfaces in Carnot groups according to their intrinsic mean curvature, with applications to the modeling of the human vision.

* We proved existence results for the mean curvature equation in the hyperbolic plane, as well as symmetry results in terms of Levi mean curvature equation in real hypersurfaces of $C^{n+1}$.

* We proved the existence of non-smooth viscosity solutions to the Levi-curvature equations with smooth prescribed curvature.

* A new notion of Maxwell's equations in groups was introduced by some members of the project, relying on a notion of intrinsic complex of differential forms introduced by M. Rumin. Basically, this complex keeps into account the lack of commutativity of the generating vector fields in a Carnot group. It has been intriguing to discover that intrinsic Maxwell's equations yield a new class of "wave equations'', with new unexpected properties that reflect the group structure and its stratification. They are, in other words, another facet of the phenomena that appear in geometric measure theory as "dimension jumps''. We have been able to show that these equations can be viewed as limits of “real” Maxwell’s equations in the matter, in presence of strong anisotropies of its magnetic and electric properties. They are also deeply connected with the notion of quasiconformal maps in a group.

* We studied Lipschitz homotopy groups of the Heisenberg group with applications to the problem of density of Lipschitz mappings in the class of Sobolev mappings into the Heisenberg group.

* We constructed an approximately differentiable homeomorphism that preserves orientation, but whose Jacobian equals $-1$ a.e. Such a homeomorphism can be obtained as a limit of uniformly convergent diffeomorphisms.

* We introduced a new notion of convexity in Carnot groups. This subject is of crucial relevance both for its applications to geometric measure theory and to the general curvature problem. In particular, a correct notion of convexity is strictly connected with the properties of the Monge-Ampère equation (also studied by participants to the project) and to the applications to ABP inequality and regularity problems for nondivergence form pde’s in Carnot groups.

* Some of the participants to the project showed that in the Euclidean setting there is a deep connection between fractional Laplacian and generalized curvatures. In the last year, the same members of the project attacked the study of fraction Laplacians in Carnot groups, aiming to find an alternative approach to the problem of general curvatures in groups.

Some of the problems raised at the beginning of the project are still open:

* We are far from having developed a full theory of currents in Carnot groups and of related isoperimetric inequality, basically because of the lack of a theory of intrinsic Lipschitz maps in higher codimension, in spite of the contribution in codimension 1 obtained by the research group. A promising interesting recent remark is the relationship between intrinsic graphs and Rumin’s differential forms.

* In spite of some results on convex sets and non-divergence form equations in Carnot groups (ABP inequalities for non-divergence form equations are the pde's counterpart of the notion of curvature), a big issue is still open: "what is the Gauss curvature in a Carnot group?'' We are studying a possible approach to this crucial question that can be given by means of the so called Steiner formula.

During the duration of the project, the members of the different teams produced about 110 papers, that have been either published or accepted by high level international scientific journals (most of them belonging to the top 33% of the ISI category "Mathematics", according the Impact Factor). A full list of these papers can be found in the project website and in the attached list of publications and abstracts.

The researchers of the projects gave a large number of talks in conferences through the world, and organized several meetings and schools that are listed in the website (http://cg-dice.dm.unibo.it/).