On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold

TL;DR

This paper proves that area-minimizing submanifolds with fractal singular sets exist on nearly all smooth manifolds, advancing the understanding of singularities in geometric measure theory.

Contribution

It extends previous work by demonstrating the existence of such submanifolds on almost any smooth manifold, broadening the scope of Almgren's conjecture.

Findings

Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.

Construction methods applicable to a wide class of manifolds.

Progress towards resolving Almgren's conjecture.

Abstract

This paper is the second in a two-part solution to Almgren's conjecture on the existence of area-minimizing submanifolds with fractal singular sets. In part one, we construct area-minimizing submanifolds with fractal singular sets on certain special manifolds. Here we continue our work and show that area-minimizing submanifolds with fractal singular sets exist on almost any smooth manifold.