Minimal laminations with prescribed convex curvature blowup

TL;DR

This paper constructs minimal laminations with specific singularities and curvature blowup rates, demonstrating that all rates between quadratic and quartic are achievable, thus generalizing previous results.

Contribution

It introduces a method to prescribe both singularities and curvature blowup rates in minimal laminations, expanding the understanding of their geometric behavior.

Findings

All curvature blowup rates between quadratic and quartic are realizable.

The construction uses perturbation techniques and PDE methods.

The work generalizes earlier results by the author and Hoffman-White.

Abstract

We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our result generalizes an earlier result of the author and of Hoffman and White.