The Concavity of the Gaussian Curvature of the convex level sets of minimal surface with respect to the height

TL;DR

This paper investigates the Gaussian curvature of convex level sets of minimal surfaces, proving its concavity with respect to height and analyzing the Catenoid as a near-sharp case.

Contribution

It introduces an auxiliary function to study curvature, establishing its concavity and providing insights into minimal surfaces like the Catenoid.

Findings

Gaussian curvature of level sets is concave in height

Auxiliary function is nearly sharp for the Catenoid

Provides new tools for analyzing minimal surface curvature

Abstract

For the minimal graph with strict convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal surface while this auxiliary function is almost sharp when the minimal surface is the Catenoid.