The Minkowski problem in Heisenberg groups

TL;DR

This paper introduces a sub-Riemannian Minkowski problem in Heisenberg groups, extending classical Euclidean concepts to a non-Abelian setting, and provides a positive solution using variational methods.

Contribution

It formulates the first sub-Riemannian Minkowski problem in Heisenberg groups and offers a novel solution approach.

Findings

Successfully formulated the sub-Riemannian Minkowski problem in Heisenberg groups

Provided a positive existence result for the problem using variational methods

Extended Minkowski theory to a new non-Euclidean geometric setting

Abstract

As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.