A global characterization of tubed surfaces in $\mathbb{C}^2$

Let $M^3\subset \mathbb{C}^2$ be a three times differentiable real hypersurface. The Levi form of $M$ transforms under biholomorphism, and when restricted to the complex tangent space, the skew-Hermitian part of the second fundamental form transforms under Möbius transformations. The surfaces for which these forms are constant multiples of each other were identified in previous work, provided the constant is not unimodular. Here it is proved that if the surface is assumed to be complete and if the constant is unimodular, then the surface is tubed over a strongly convex curve. The converse statement is true, too, and is easily proved.