Rational approximation and Lagrangian inclusions

Abstract

We show that any real compact surface $S$, except the sphere $S^2$ and the projective plane $\mathbb{R}{P}_2$, admits a pair of smooth complex-valued functions $f_1,f_2$ with the property that any continuous complex-valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions on ${\mathbb{C}}^2$.