Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

TL;DR

This paper establishes sharp comparison and maximum principles for H-convex functions in the Heisenberg group using horizontal normal mapping and degree theory, providing new insights into boundary behavior and answering open questions.

Contribution

It introduces a novel approach based on horizontal normal mapping and degree theory to prove comparison and maximum principles for H-convex functions in the Heisenberg group.

Findings

Proved a comparison principle for H-convex functions.

Established an Aleksandrov-type maximum principle.

Provided examples demonstrating sharpness of results.

Abstract

In this paper we solve a problem raised by Guti\'errez and Montanari about comparison principles for $H-$convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous $H-$convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.