Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata

Abstract

This paper is a continuation of our work on a conjecture of Almgren on area-minimizing surfaces with fractal singular sets. First, we prove that area-minimizing surfaces with fractal singular sets are prevalent on the homology level on general manifolds. They exist even in metrics arbitrarily close to the flat metric on a torus. Second, we partially determine the moduli space of area-minimizing currents near the ones with fractal singular set that we constructed in this and previous work. Around strata of codimension at least three, the fractal self-intersections deform into transverse immersions under generic perturbations of metric. The result is sharp and we construct concrete examples to show that our fractal singularities do not necessarily completely dissolve under generic perturbations. Finally, we improve our previous results to give examples with different strata limiting to each other and refine the first stratum to fractal subsets of finite combinatorial graphs.