A compact set $K \subset \mathbb{C}^n$ is said to be rationally convex if its complement is a union of complex hypersurfaces. Rational convexity has an important approximation-theoretic consequence: every function holomorphic on a neighborhood of $K$ can be uniformly approximated on $K$ by rational functions with poles off $K$. In the 1990s, Duval ($n=2$) and Duval–Sibony ($n\ge2$) established a remarkable symplectic characterization of rational convexity for compact totally real submanifolds of $\mathbb{C}^n$. They proved that such a submanifold is rationally convex if and only if it is isotropic with respect to some K"ahler form on $\mathbb{C}^n$.

In this talk, we consider $n$-dimensional totally real immersed submanifolds of $\mathbb{C}^n$ whose only self-intersections consist of finitely many transverse double points. For such a submanifold $S$, Gayet (2000) proved that if $S$ is Lagrangian with respect to some K"ahler form on $\mathbb{C}^n$, then $S$ is rationally convex. Later, Mitrea (2020) showed that the converse fails in general and established a partial converse to Gayet’s theorem.

We will first give a brief overview of the proof of Gayet’s theorem. We will then present a new result which provides a complete symplectic characterization of rational convexity for immersed totally real submanifolds with finitely many transverse double points by identifying the precise condition required at the double points.