arXiv research

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

The study examines optimal synthesis in a radially symmetric Grushin space with conditions on the weight function.

Classifies quantum particle behavior on a special cylinder.

We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of…

We employ Grushin jets which are adapted to the geometry of Grushin-type spaces to obtain the existence-uniqueness of viscosity solutions to the $\infty(x)$-Laplace equation in Grushin-type spaces. Due to the differences between Euclidean jets and Grushin jets, the Euclidean method of proof is not valid in this environ…

Computed distortion coefficients for the α-Grushin plane.

Study on quantum particle evolution on Grushin cylinder, embedding in R^3.

New cylindrical solutions found for Grushin-type problem.

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…

Study geodesics on Grushin spaces, proving upper bounds on conjugate times.

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

We address the double bubble problem for the anisotropic Grushin perimeter $P_α$, $α\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the case of two equal volumes. We assume that the contact interface between the bubbles lays on either the vertical or the horizontal axis. Since no regularity theory is available i…

New sub-Riemannian spaces with boundary meet curvature-dimension condition.

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

The study proves sub-Riemannian manifolds cannot satisfy $\mathrm{CD}$ conditions unless they are Riemannian.

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…

Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.

Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq κ$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces…

We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of…

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometric…

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

Study on heat flow across two half-lines with special boundary conditions.

We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing i…

New Weyl's laws discovered for compact spaces with Ricci curvature bounds.

Paper studies viscosity solutions in unique Martinet spaces.

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, …

We find the fundamental solution to the p-Laplace equation in a class of Hörmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space…

New theorem on embedding Moebius bands in 3D space.

A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Fin…

We study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnorma…

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As …

Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.

For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces…

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…

We obtain the best known quantitative estimates for the $L^p$-Poincaré and log-Sobolev inequalities on domains in various sub-Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H-type Carnot groups and the Heisenberg groups, corank $1$ Carnot groups, the Grushin plane, and various H…

This thesis surveys various metrics on Riemann surface spaces.

Proves existence and uniqueness of weighted metrics for smooth spaces.

New Finsler metrics constructed from $GDW$-metrics.

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|β\|_α>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…

We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…

New metric defined for bounded symmetric domains.

Introduces Finslerian convolution metrics and their properties.

Survey of recent metric geometry in Kähler metrics space.

Study on geodesics of Finsler metrics derived from Riemannian metrics.