Curvature bounds on length-minimizing discs

TL;DR

This paper demonstrates that length-minimizing disks inherit the upper curvature bounds of their target space, extending this property to harmonic and ruled disks within the same curvature constraints.

Contribution

It establishes that length-minimizing, harmonic, and ruled disks inherit the upper curvature bounds from their ambient space, providing new insights into geometric properties of minimal surfaces.

Findings

Length-minimizing disks inherit the target's curvature bounds.

Harmonic disks inherit the upper curvature bounds.

Ruled disks also inherit the upper curvature bounds.

Abstract

We show that a length-minimizing disk inherites the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.