Polyhedral approximation by Lagrangian and isotropic tori

We prove that every smoothly immersed $2$-torus of $\mathbb{R}^4$ can be approximated, in the $C^0$-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the $C^1$-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic $2$-tori of $\mathbb{R}^{2n}$.

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Published 26 April 2023