Stacking Disorder In Periodic Minimal Surfaces

We construct one-parameter families of nonperiodic embedded minimal surfaces of infinite genus in T x R, where T denotes a flat 2-tori. Each of our families converges to a foliation of T x R by T. These surfaces then lift to minimal surfaces in R-3 that are periodic in horizontal directions but not periodic in the vertical direction. In the language of crystallography, our construction can be interpreted as disordered stacking of layers of periodically arranged catenoid necks. Limit positions of the necks are governed by equations that appear, surprisingly, in recent studies on the mean field equation and the Painleve VI equation. This helps us to obtain a rich variety of disordered minimal surfaces. Our work is motivated by experimental observations of twinning defects in periodic minimal surfaces, which we reproduce as special cases of stacking disorder.