Rationally Convex Surfaces with hyperbolic complex tangencies

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by handle-bodies, and those that do not have any compact Riemann surfaces attached at all. The fillable examples all live in the round sphere and are unknotted, while the non-fillable examples can moreover be produced in several different smooth isotopy classes.
Comment: 26 pages, 11 figures