Polyhedral approximation by Lagrangian and isotropic tori

TL;DR

This paper demonstrates that smooth Lagrangian and isotropic 2-tori in Euclidean spaces can be approximated by polyhedral versions in various smoothness senses, advancing geometric approximation methods.

Contribution

It introduces new approximation results for smooth Lagrangian and isotropic tori by polyhedral tori in different smoothness categories.

Findings

Approximation of smooth Lagrangian tori by polyhedral Lagrangian tori in C0-sense.

Approximation of embedded Lagrangian tori in C1-sense.

Extension of approximation results to isotropic tori in higher dimensions.

Abstract

We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-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 C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $\mathbb{R}^{2n}$.