Stability of p-area minimizing surfaces in the Heisenberg group

TL;DR

This paper develops a quantitative stability theory for p-area minimizing surfaces with prescribed nonzero p-mean curvature in the Heisenberg group, using duality and geometric analysis.

Contribution

It introduces a novel stability framework for minimizers with prescribed mean curvature, including explicit rates and stability estimates in the Heisenberg group.

Findings

Established $L^1$ stability of minimizers under geometric assumptions.

Derived $W^{1,1}$ stability estimates with explicit rates in low dimensions.

Provided the first quantitative stability results for nonzero p-mean curvature graphs.

Abstract

We study the stability of minimizers of weighted $p$-area functionals associated with prescribed $p$-mean curvature surfaces in the Heisenberg group. While existence and uniqueness results are well established, quantitative stability with respect to perturbations of the mean curvature $H$ remains largely unexplored in the nonzero-$H$ regime. Using a Rockafellar--Fenchel duality framework, we identify a unique underlying vector field associated with each minimizer and prove its stability under perturbations of $H$. This yields quantitative control of the direction field of the horizontal gradient. Building on this structure, we establish $L^1$ stability of admissible minimizers under natural geometric assumptions on level sets. In dimensions two and three, we also derive $W^{1,1}$ stability estimates under additional regularity and structural hypotheses, with explicit rates in terms of $\|H-\tilde H\|_{L^\infty}$. Our results provide the first quantitative stability theory for $p$-area minimizing graphs with prescribed nonzero $p$-mean curvature, even in the unweighted case. Numerical simulations are included to illustrate the robustness of the theoretical results.