Level Curves of Minimal Graphs

TL;DR

This paper establishes a curvature inequality for level sets of minimal graphs with zero boundary conditions and demonstrates that boundary concavity implies all level sets are concave.

Contribution

It introduces a new inequality relating curvature of level sets to boundary conditions and proves concavity of level sets under boundary concavity.

Findings

Curvature inequality for minimal graph level sets

Concavity of level sets when boundary is concave

All level sets are concave if boundary is concave

Abstract

We prove an inequality for the curvature of level sets of minimal graphs having vanishing boundary values and show that if the boundary is concave, then all the level sets are concave.