Minimal tori in $\mathbb{R}^4$

Abstract: We describe tools for the study of minimal surfaces in $\mathbb{R}^4$; some are classical (the Gauss maps) and some are newer (the link/braid/writhe at infinity). Then we look for complete proper non holomorphic minimal tori with total curvature $-8\pi$ and a single end immersed in $\mathbb{R}^4$. We translate the problem into a system of $10$ quadratic or linear equations in $11$ real variables with coefficients in terms of the Weierstrass function $\wp$ and give explicit solutions for these equations if $T$ is a rectangular torus. For the square torus, we have a complete answer with a unique family of solutions generalizing the Chen-Gackstetter torus in $\mathbb{R}^3$. On the other hand, we show that there is no solution on the equianharmonic torus.