Gaussian curvature conjecture for minimal graphs

Abstract

In this paper, we solve the longstanding Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. The conjecture asserts that for any minimal graph above the unit disk, the Gaussian curvature at the point directly above the origin satisfies the sharp inequality \( |\mathcal{K}| < \frac{π^2}{2} \). We first reduce the conjecture to the problem of estimating the Gaussian curvature of certain Scherk-type minimal surfaces defined over bicentric quadrilaterals inscribed in the unit disk, containing the origin. We then provide a sharp estimate for the Gaussian curvature of these minimal surfaces at the point above the origin. Our proof employs complex-analytic methods, as the minimal surfaces in question allow a conformal harmonic parameterization.