Blow-ups of minimal surfaces in the Heisenberg group

TL;DR

This paper revises Monti's analysis of blow-ups of minimal surfaces in the Heisenberg group, correcting the PDE for the limit function and clarifying its properties in the context of geometric measure theory.

Contribution

The paper corrects the PDE derived for the limit function in the blow-up analysis of minimal surfaces in the Heisenberg group, providing a more accurate characterization.

Findings

The horizontal Laplacian of the limit function is independent of the coordinate y_1.

The corrected PDE is satisfied weakly by the limit function.

Monti's original PDE for the limit function was incorrect.

Abstract

In this paper, we revise Monti's results on the blow-ups of H-perimeter minimizing sets in $\mathbb{H}^n$. Monti demonstrated that the Lipschitz approximation of the blow-up, after rescaling by the square root of the excess, converges to a limit function for $n \ge 2$. However, the partial differential equation he derived for this limit function $\varphi$ through contact variation is incorrect. Instead, the correct equation is that the horizontal Laplacian of the limit function $\varphi$ is independent of the coordinate $y_1$ and solves equation 1 weakly.