# Jacobian determinant inequality on corank 1 Carnot groups with applications

## TL;DR

This paper proves a Jacobian determinant inequality on corank 1 Carnot groups, bridging Euclidean and sub-Riemannian geometries, and applies it to derive fundamental inequalities like entropy and Brunn-Minkowski.

## Contribution

It introduces a weighted Jacobian inequality in corank 1 Carnot groups that handles abnormal geodesics, extending optimal transport theory to this setting.

## Key findings

- Established a Jacobian determinant inequality for corank 1 Carnot groups.
- Derived entropy, Brunn-Minkowski, and Borell-Brascamp-Lieb inequalities.
- Connected sub-Riemannian and Euclidean optimal transport frameworks.

## Abstract

We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups.