The Fujimoto Conjecture via Total Positivity

H. Fujimoto showed that for a complete minimal surface in $\mathbb{R}^m$, if the Gauss map is non-degenerate, then it omits at most $\frac{m(m + 1)}{2}$ hyperplanes in the complex projective space $\mathbb{P}^{m - 1}$ in general position, and that the number $\frac{m(m + 1)}{2}$ is best possible for all odd integers $m \geq 3$ and for even integers with $4 \leq m \leq 16$. In this paper, we prove that the number $\frac{m(m + 1)}{2}$ is also best possible for all even integers $m \geq 4$, as conjectured by Fujimoto. The main tool is a special planar network $(\Gamma_0, \omega)$ in the theory of positive matrices.