Rational approximation and Lagrangian inclusions

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb RP_2$, 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 in $\mathbb C^2$.

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