Some special vector fields and their dual fields (Q1611382)

Let \(\Omega\) be a domain in \(\mathbb{R}^n\) with smooth boundary \(\Sigma\). The ``capillary problem'' asks for a function \(z:\overline\Omega \to \mathbb{R}\) such that its graph \(S\) has constant mean curvature \(H\) and \(S\) meets the cylinder \(\Sigma\times\mathbb{R}\) with a prescribed angle \(\gamma\). A function \(z\) is a solution of the problem if and only if the vector field \(Z:=\text{grad} z/ \sqrt{1+ \|\text{grad} z\|^2}\) satisfies \(\text{div} Z |_\Omega=nH\) and \(\langle Z,\nu\rangle |_\Sigma= \cos\gamma\), where \(\nu\) denotes the outer normal of \(\Sigma\). In the article the vector field \(\overline Z\) on the boundary \(\Sigma\) is considered, which is obtained by projecting \(Z_p\) orthogonally onto \(T_p\Sigma\). For instance, the author asks under which conditions \(\overline Z\) is a Killing field resp. harmonic. Unfortunately he assumes that ``always'' \(\Sigma\) has a parametrization by lines of principal curvature.