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 $\int |\nabla u+\vec{F}| + Hu$. 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.

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