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 2 satisfies the sharp inequality |K | < π²/<inf>2</inf>. 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.