Sequences of embedded minimal disks whose curvatures blow up on a prescribed subset of a line

TL;DR

This paper constructs sequences of embedded minimal disks in 3D space with curvature blow-up exactly on a prescribed closed subset of a line segment, demonstrating precise control over curvature behavior.

Contribution

It introduces a method to produce minimal disks with curvature blow-up on any given closed subset of a line segment in Euclidean space.

Findings

Minimal disks can be constructed with curvature blow-up on arbitrary closed sets.

Curvature remains bounded away from the prescribed subset.

The construction is proper and embedded in a solid cylinder.

Abstract

For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.