Rationally Convex Surfaces with hyperbolic complex tangencies

TL;DR

This paper constructs the first examples of rationally convex surfaces in the complex plane exhibiting hyperbolic complex tangencies, distinguishing between fillable and non-fillable types with different topological properties.

Contribution

It provides the first known examples of such surfaces, including both fillable and non-fillable types, expanding understanding of complex tangencies in rationally convex surfaces.

Findings

Fillable examples are unknotted and reside in the round sphere.

Non-fillable examples can be produced in various smooth isotopy classes.

The paper demonstrates the existence of rationally convex surfaces with hyperbolic complex tangencies.

Abstract

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.