arXiv research

Given a real-valued function $c$ defined on the cartesian product of a generic Carnot group $\G$ and the first layer $V_1$ of its Lie algebra, we introduce a notion of $c$ horizontal convex ($c$ H-convex) function on $\G$ as the supremum of a suitable family of affine functions; this family is defined pointwisely, and …

Formula calculates equilibria of convex bodies based on evolute winding number.

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations …

Paper studies how certain hypersurfaces collapse in Hilbert space.

Characterizes Kähler-Berwald metrics on complex manifolds.

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the bou…

Study rigid motions in 3D-Heisenberg group to define and relate geometric probabilities.

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.

Study on shapes in Heisenberg group with convex body norms.

Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given…

Study geodesics in Heisenberg groups with sub-Finsler metrics.

Step 2 Carnot groups can approximate horizontal curves without error.

Defines conformal submersion with horizontal distribution and provides necessary conditions for its existence.

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.

We describe explicit horizontal open books on some Seifert fibered 3--manifolds. We show that the contact structures compatible with these horizontal open books are Stein fillable and horizontal as well. Moreover we draw surgery diagrams for some of these contact structures.

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…

Study H-type foliations in sub-Riemannian geometry.

The study finds conditions for horizontal envelopes in a 3D-Heisenberg group.

The paper studies critical points of horizontal energy functional in Riemannian foliations.

Study horizontal discs in fat distributions, proving their existence.

The study develops inequalities for Riemannian foliations without bundle-like assumptions.

Fundamental weight systems identified as quantum states.

Affine and conformal submersions with horizontal distribution are studied in statistical manifolds.

Study shows horizontal diameter of unit spheres is π for specific foliations.

The complexity of horizontality in twistor spaces on tori is infinite.

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…

Smooth 4-manifolds have simple horizontal decompositions.

We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.

New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.

Study on curvatures of surfaces in specific Lie groups.

In this paper, we study the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds, where "almost free" means that the stabilizers of the group action are finite. First w…

Examples of area-minimizing graphs with low regularity in a specific group.

Study examines how hyperbolic metrics change over time.

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a conseque…

Paper proves unique Finsler connections for scalar forms.

The study examines constant mean curvature tubes around geodesics in specific 3-manifolds.

Method calculates $\mathbb{Z}_2$-Thurston norm for Seifert 3-manifolds using pseudo-horizontal surfaces.

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation $H+σ=0$, where $H$ is analogous to the…

The paper characterizes gauge balls in the Heisenberg group by their curvature.

Extends Masur's divergence theorem to complex tori and Kummer surfaces.

We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally …

We define and study the invariant linear and nonlinear horizontal double complexes of a local Lie group.

We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.

Finite intersection numbers between horizontal foliations of quadratic differentials.

Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.

We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.