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, we prove that \(N_{\vec {F}}(u) = N_{ \vec {F}}(v)\) implies \(\nabla u = \nabla v\) 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.