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

We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$\mathrm{.}If$u$and$v$aretwominimizersforthegeneralized$p$-areasatisfyingthesameDirichletboundarycondition,thenwecanonlyget$N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ (on the nonsingular set) where $N_{\vec{F}}(w)$ $:=$ $\frac{\mathrm{\nabla }w+\vec{F}}{\mathrm{\mid }\mathrm{\nabla }w+\vec{F}\mathrm{\mid }}\mathrm{.}$ To conclude $u$ $=$ $v$ (or $\mathrm{\nabla }u$ $=$ $\mathrm{\nabla }v)$, 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 $\mathrm{\nabla }u$ $=$ $\mathrm{\nabla }v$ in dimension $\ge$ 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.