Rational approximation and Lagrangian inclusions (Q2406617)

The authors prove a nice two-dimensional generalization of a classical result that any continuous function on \(S^1\subset\mathbb{C}\) can be uniformly approximated by rational functions of \(e^{i\theta}\) with the poles off \(S^1\). The result answers (affirmatively) a question of Nemirovsky. Let \(S\) be a real compact surface, other than \(S^2\) or \(\mathbb{R}P^2\), then there is a pair of smooth complex valued functions \(f_1\) and \(f_2\) on \(S\) such that any continuous function \(f\) on \(S\) is a uniform limit of \(R_n(f_1,f_2)\), where the \(R_n\) are rational and have denominators not vanishing on \(S\). If \(S\) is a \(2\)-torus one can take \(e^{i\theta_1}\), \(e^{i\theta_2}\) as \(f_1\), \(f_2\). When \(S\) is \(S^2\) or \(\mathbb{R}P^2\) the convergence can only be guaranteed to \(f\circ\tau\), where \(\tau\) is an \(f\)-dependent rotation for \(S^2\) and an \(f\)-dependent smooth diffeomorphism for \(\mathbb{R}P^2\). While the statement is elementary the proof is not. The functions are constructed as the coordinates of a singular Lagrangian embedding of \(S\) into \(\mathbb{C}^2\). By a result of Givental one can always find a Lagrangian inclusion with double intersection points, and the authors show that one can even find such an embedding with singularities being open Whitney umbrellas, unless \(S\) is \(S^2\) or \(\mathbb{R}P^2\). The last two can only have inclusions, spoiling the approximation. Moreover, even though no Lagrangian embedding of \(S\) can have a polynomially convex image the authors prove along the way that there always exists an inclusion whose image is rationally convex (i.e., for any \(z\not\in S\) there is an algebraic hypersurface passing through \(z\) and avoiding \(S\)). This further confirms a close relationships between the Lagrangian property and rational convexity first observed by Duval.