# Stable minimal graphs in Heisenberg group $\mathbb{H}^n$

## TL;DR

This paper proves that strictly stable minimal graphs in the Heisenberg group are locally area-minimizing and establishes the existence and uniqueness of smooth minimal graphs with small boundary data.

## Contribution

It demonstrates the local area-minimizing property of stable minimal graphs and proves existence and uniqueness of smooth minimal graphs with small boundary conditions in the Heisenberg group.

## Key findings

- Stable minimal graphs are locally area-minimizing.
- Existence of smooth minimal graphs with small boundary data.
- Uniqueness of these minimal graphs.

## Abstract

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum.