Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group

TL;DR

This paper investigates the uniqueness of generalized p-area minimizers in the Heisenberg group, establishing conditions under which the horizontal normal determines the gradient of minimizers and exploring integrability conditions for such normals.

Contribution

It proves that under certain conditions, the horizontal normal uniquely determines the gradient of minimizers in dimensions three and higher, and develops an integrability criterion for horizontal normals.

Findings

N_F(u) = N_F(v) implies ∇u = ∇v in dimension ≥ 3 under specific conditions.

In dimension 2, the same implication does not hold.

Derived a Codazzi-like equation as an integrability condition for horizontal normals.

Abstract

We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ (on the nonsingular set) where $N_{\vec{F}}(w)$ $:=$ $\frac{\nabla w+\vec{F}}{|\nabla w+\vec{F}|}.$ To conclude $u$ $=$ $v$ (or $\nabla u$ $=$ $\nabla v)$, it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, for a generalized area functional including $p$-area, we prove that $N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ implies $\nabla u$ $=$ $\nabla v$ in dimension $\geq $ 3 under some rank condition on derivatives of $\vec{F}$ or the nonintegrability condition of contact form associated to $u$ or $v$. Note that in dimension 2 ($n=1),$ the above statement is no longer true. Inspired by an equation for the horizontal normal $N_{\vec{F}}(u),$ we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.