The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks

TL;DR

This paper initiates a series aiming to comprehensively describe embedded minimal surfaces of fixed genus in 3-manifolds, focusing on local disk structures and estimates off the axis in Euclidean space.

Contribution

It introduces new estimates for embedded minimal disks in Euclidean space, advancing understanding of their local structure and behavior.

Findings

Develops estimates off the axis for minimal disks

Provides foundational results for classifying minimal surfaces

Sets the stage for a complete description of minimal surface moduli

Abstract

This 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).