The local picture theorem on the scale of topology

TL;DR

This paper establishes a local geometric structure theorem for minimal surfaces with zero injectivity radius in three-manifolds, introducing a new limit object called a minimal parking garage structure on c^3.

Contribution

It provides a new descriptive theorem on the extrinsic geometry of minimal surfaces and develops the theory of minimal parking garage structures as limit objects.

Findings

Characterization of minimal surfaces near points of almost-minimal injectivity radius

Introduction and development of minimal parking garage structures on c^3

A limit object describing the local geometry of minimal surfaces

Abstract

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.