Gaussian curvature conjecture for minimal graphs

TL;DR

This paper proves the Gaussian curvature conjecture for minimal graphs over the unit disk, establishing a sharp inequality for the curvature at the origin using complex analysis and minimal surface estimates.

Contribution

It resolves the longstanding conjecture by reducing it to Scherk-type surfaces and providing sharp curvature estimates through complex-analytic methods.

Findings

Confirmed the sharp inequality |K| < A0^2/2 for minimal graphs.

Reduced the conjecture to curvature estimates of Scherk-type surfaces.

Provided a novel application of complex analysis in minimal surface theory.

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 \( |\mathcal{K}| < \frac{\pi^2}{2} \). 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.