2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65208This paper is the first 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 Riemannian 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 $\RR^3$ (with the flat metric).Figures added to existing preprintAnalysis of PDEsDifferential Geometrytext