Zero-curvature point of minimal graphs

TL;DR

This paper investigates the extremal properties of the second derivative of Gaussian curvature at the center of minimal graphs, revealing that certain Scherk-type surfaces over a hexagon are extremal under specific conditions.

Contribution

It extends Finn and Osserman's classical result by characterizing extremal minimal surfaces with zero curvature and horizontal tangent plane at the center, involving Scherk-type surfaces over a hexagon.

Findings

Extremals are Scherk-type minimal surfaces over a hexagon.

The second derivative of Gaussian curvature is extremized by these surfaces.

Results generalize classical curvature extremal problems for minimal graphs.

Abstract

Motivated by a classical result of Finn and Osserman (1964), who proved that the Scherk surface over the square inscribed in the unit disk is extremal for the Gaussian curvature of the point $O$ (so-called \emph{centre}) of the minimal graphs above the center $0$ of unit unit disk, provided the tangent plane is horizontal, we ask and answer to the question concerned the extremal of "second derivative" of the Gaussian curvature of such graphs provided that its curvature at $O$ is zero. We prove that the extremals are certain Scherk type minimal surfaces over the regular hexagon inscribed in the unit disk, provided that the Gaussian curvature vanishes and the tangent plane is horizontal at the centre.