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This is a Quantitative Study study.

October 19, 2025Open Access

Minimal tori in R⁴

Key Points

Unique families of explicit solutions exist for square tori, generalizing earlier results in R^3.
The study establishes a system of 10 equations in 11 variables related to the Weierstrass function.
Complete non holomorphic minimal tori with total curvature -8π are shown to exist within R^4.
No solutions are found for the equianharmonic torus, highlighting constraints in the model.

Abstract

We describe tools for the study of minimal surfaces in R⁴; 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 -8π and a single end immersed in R⁴. 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 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 R³. On the other hand, we show that there is no solution on the equianharmonic torus.

Minimal tori in R⁴

Key Points

Abstract

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