Related papers: Jacobian determinant inequality on corank 1 Carnot…

We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is…

This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension…

The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Amp\`ere type known as generated Jacobian equations. These equations, whose general existence theory has been recently…

In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math., 2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Mass Transportation, raising the question whether their approach is powerful enough to…

Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager [{\it Invent. Math.,} 2001], we derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship…

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the…

In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We derive the Hamiltonian geodesic equations for such structures, and we provide…

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first…

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli-Kohn-Nirenberg type, where the weights involved are powers of the Carnot-Caratheodory distance associated with a fixed system of vector fields…

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the…

Equations of motion for general gravitational connection and orthonormal coframe from the Einstein-Hilbert type action are derived. Our formulation does not fix coframe to be tangential to spatial section hence Lorentz group is still…

For the result on 1-quasiconformal maps, see the paper by Cowling and Ottazzi. The result on quasiconformal maps on Carnot groups with reducible first layer will appear in a forthcoming paper by Enrico Le Donne and Xiangdong Xie.

The optimal transport map between the standard Gaussian measure and an $\alpha$-strongly log-concave probability measure is $\alpha^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two…

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a…

We prove that the Benamou-Brenier formulation of the Optimal Transport problem and the Kantorovich formulation are equivalent on a sub-Riemannian connected and complete manifold $M$ without boundary and with no non-trivial abnormal…

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the…

We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a first consequence, we show in full gener- ality that the log-Sobolev inequality is equivalent to an hypercontractivity…

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i. e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over…

We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weighted versions of these inequalities. Our approach generalizes…

Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable. Some are not. The space of k-jets for real-valued…