Discrete geometry and isotropic surfaces

TL;DR

This paper studies isotropic immersions of tori into Euclidean spaces, showing they can be approximated by piecewise linear maps and introducing a numerical flow to generate examples of Lagrangian tori.

Contribution

It demonstrates approximation of smooth isotropic immersions by piecewise linear maps and develops a numerical flow for visualizing Lagrangian tori in four-dimensional space.

Findings

Approximation of smooth isotropic immersions by piecewise linear maps.

Development of a numerical flow to generate Lagrangian tori.

Implementation of the DMMF program for real-time visualization.

Abstract

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.