arXiv research

Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.

Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.

Study shows only hyperplanes in Heisenberg groups have zero curvature.

Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.

Paper estimates curvature of minimal surfaces in a specific geometric space.

Harmonic maps studied in sub-Riemannian geometry for Lie groups.

Study local control in a 7D quaternionic Heisenberg group.

Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.

Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.

Injective X-ray transform on Heisenberg group for regular functions.

The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.

Sharp stability of isometries on Heisenberg group proven.

Study shows stable graphs in Heisenberg group are essentially planes.

Integrability of mean curvature near degenerate points in Heisenberg group.

Study geodesic curvature in Heisenberg group, interpreting it as distance correction.

The study proves intrinsic graphs in Heisenberg group are planes if they meet certain conditions.

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curve…

Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.

The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.

Study on integrability of geodesic flows on Heisenberg group.

Proves properties of sub-Riemannian exponential map, showing it's not injective.

The affine-additive group is hyperbolic with a non-vanishing 4-capacity.

Study on curvatures of surfaces in specific Lie groups.

This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental inte…

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.

Computer program finds flower-like periodic orbits in Kepler-Heisenberg problem.

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…

New method for sampling diffusion bridges on sub-Riemannian manifolds.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solution…

Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.

Develops non-Markovian couplings for sub-Riemannian Brownian motions.

Paper proves inequalities for forms on sub-Riemannian manifolds.

Let Hn denote the (2n + 1)-dimensional (sub-Riemannian) Heisenberg group. In this note, we shall prove an integral identity (see Theorem 1.2) which generalizes a formula obtained in the Seventies by Reilly. Some first applications will be given in Section 4.

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

We investigate the notion of H-subdifferential and H-normal map of a function on the Heisenberg group, based on its sub-Riemannian structure. In particular, a characterization of the convexity of a function is given via the nonemptiness of the H-subdifferential at every point.

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

Paper shows limits of Heisenberg manifolds are flat tori.

We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…

Zero-energy orbits in the Kepler-Heisenberg problem are self-similar and stratify into three families.

In the context of sub-Riemannian Heisenberg groups Hn, n \geq 1, we shall study Isoperimetric Profiles, which are closed compact hypersurfaces having constant horizontal mean curvature, very similar to ellipsoids. Our main goal is to study the stability of Isoperimetric Profiles.

Derives sub-Riemannian Ricci curvature for various manifolds.

The paper studies metrics and geodesics on a quaternionic Heisenberg group.