Negatively Curved Minimal Surfaces in the Round Four-Sphere

Abstract: We construct smooth closed embedded minimal surfaces with everywhere negative Gaussian curvature in the unit round four-sphere. More precisely, for every sufficiently large integer $m$, our construction gives a minimal surface $Σ_m$ of genus $m<sup>2+1$ such that $\operatorname{Area}(Σ_m)\to 4π<sup>2$ as $m\to+\infty$. This result yields a complete solution to Problem 101 in Yau's 1982 list of open problems.