The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks

TL;DR

This paper investigates the local structure of embedded minimal disks in three-dimensional space, showing that high curvature regions contain nearly flat multi-valued graphs extending close to the boundary, aiding the classification of minimal surfaces.

Contribution

It establishes that high curvature points in minimal disks imply the existence of almost flat multi-valued graphs, advancing the understanding of minimal surface structure in 3-manifolds.

Findings

High curvature points contain almost flat multi-valued graphs

Multi-valued graphs extend nearly to the boundary of the disk

Provides tools for classifying embedded minimal surfaces

Abstract

This 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.