Related papers: Geometric inequalities in Carnot groups

The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost…

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result…

This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy…

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of…

We prove that if the shape of the metric unit ball in a homogeneous group enjoys a precise symmetry property, then the associated distance yields the standard form of the area formula. The result applies to some classes of smooth and…

We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the $C^0$-metric and the Hofer…

In this paper we construct infinitely many examples of a Riemannian submersion from a simple, compact Lie group $G$ with bi-invariant metric onto a smooth manifold that cannot be a quotient of $G$ by a group action. This partially addresses…

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart…

We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is computed with respect to an arbitrary homogeneous distance. Among the arguments…

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…

Quantum Graphity (QG) is a model of emergent geometry in which space is represented by a dynamical graph. The graph evolves under the action of a Hamiltonian from a high-energy pre-geometric state to a low-energy state in which geometry…

We investigate the infinitesimal invariants of an immersed submanifold $\Sigma $ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $\Sigma $. The invariants are derived from the logarithmic…

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on…

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in…

Rank-one symmetric spaces carry a solvable group model which have a generalization to a larger class of Lie groups that are one-dimensional extensions of nilpotent groups. By examining some metric properties of these symmetric spaces, we…

We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a K\"unneth formula \[ H^{p,q}(M\times N) \cong \bigoplus_{i,j} H^{i,j}(M)\otimes H^{p-i,q-j}(N) \] for flat affine manifolds $M,N$ with at least one…

We study mappings on sub-Riemannian manifolds which are quasi-regular with respect to the Carnot-Caratheodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using…

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal…

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups $\mathbb{H}^n$. Among other results we prove that measures with $(2n+1)$-density have only flat tangents and conclude the…