Minimal surface doublings and electrostatics for Schr\"odinger operators

Abstract

Twenty years ago, N. Kapouleas introduced a singular perturbation construction known as "doubling", which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. Doubling constructions have since been implemented successfully in several settings, with deep work of Kapouleas-McGrath reducing their existence theory to the problem of finding suitable families of ansatz data on the initial minimal surface. In this paper, we introduce a variational approach to the existence of minimal doublings, relating the Kapouleas-McGrath construction to the study of nondegenerate critical points for a Coulomb-type interaction energy. By analyzing the minimizers of this energy, we prove that, in a generic closed 3-manifold, every two-sided, embedded minimal surface of index one admits a sequence of minimal doublings. As a corollary, we find that a generic 3-manifold contains an infinite sequence of embedded minimal surfaces with bounded area and arbitrarily large genus, whose geometry can be described with some precision.