For square and rectangular tori, the paper gives explicit non-holomorphic minimal tori in R^4 with total curvature -8π and one end, and proves uniqueness on the square torus.
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Minimal tori in $\mathbb{R}^4$
For square and rectangular tori, the paper gives explicit non-holomorphic minimal tori in R^4 with total curvature -8π and one end, and proves uniqueness on the square torus.