2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65191This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in $\RR^3$. We show here that if the curvature of such a disk becomes large at some point, then it contains an almost flat multi-valued graph nearby that continues almost all the way to the boundary.Figures added to existing preprintAnalysis of PDEsDifferential Geometrytext