On a sharp form of curvature conjecture for minimal graphs

TL;DR

This paper advances the understanding of Gaussian curvature bounds for minimal graphs over the unit disk, providing sharp inequalities that generalize classical results and constructing examples with prescribed normals.

Contribution

It improves the sharp curvature inequality for minimal graphs, extends results to arbitrary normals, and constructs minimal graphs approaching Scherk-type surface curvatures.

Findings

Proved the sharp inequality | K| < π^2/2 for minimal graphs.

Constructed minimal graphs with prescribed normals approaching Scherk-type surface curvatures.

Extended classical results to more general normal vectors.

Abstract

Recently, the author and Melentijevi\'c resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant \( c_0 \) is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant \( c_1 \). In addition, we show that for any prescribed unit normal vector \( \mathbf{n} \), there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is \( (0,0,1) \).