Curvature of minimal graphs

TL;DR

This paper investigates the Gaussian curvature of minimal graphs over the unit disk, reducing the conjecture to specific Scherk-type surfaces, and provides improved and optimal curvature estimates under symmetric conditions.

Contribution

It reduces the Gaussian curvature conjecture to estimating Scherk-type minimal surfaces and extends classical results with new optimal curvature bounds under symmetry.

Findings

Improved upper estimates of Gaussian curvature at the disk center.

Derived an optimal curvature estimate for symmetric points.

Constructed a family of Scherk's type minimal graphs for comparison.

Abstract

We consider the Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk's type minimal surfaces over a quadrilateral inscribed in the unit disk containing the origin inside. As an application we improve so far the obtained upper estimates of Gaussian curvature at the point above the center. Further we obtain an optimal estimate of the Gaussian curvature at the point $\mathbf{w}$ over the center of the disk, provided $\mathbf{w}$ satisfies certain "symmetric" conditions. The result extends a classical result of Finn and Osserman in 1964. In order to do so, we construct a certain family $S^t$, $t\in[t_\circ, \pi/2]$ of Scherk's type minimal graphs over the isosceles trapezoid inscribed in the unit disk. Then we compare the Gaussian curvature of the graph $S$ with that of $S^t$ at the point $\mathbf{w}$ over the center of the disk.