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

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.