Existence and structure of P-area minimizing surfaces in the Heisenberg group

TL;DR

This paper investigates the existence and structure of P-area minimizing surfaces in the Heisenberg group, introducing a new approach that advances understanding of their existence under boundary conditions.

Contribution

It introduces a novel vector field characterization and a barrier condition criterion, providing a new method for establishing the existence of P-area minimizing surfaces.

Findings

Existence of a vector field N characterizing P-area minimizing surfaces.

Barrier condition on boundary ensures existence of such surfaces.

Progress in understanding the structure of P-area minimizing surfaces.

Abstract

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of $P$-area minimizing surfaces. This vector field exists even if there is no $P$-area minimizing surface satisfying the prescribed boundary conditions. We prove that if $\partial \Omega$ satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and makes major progress in understanding existence of $P$-area minimizing surfaces.