## Journal of Differential Geometry

### The local picture theorem on the scale of topology

#### 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.

#### Note

First author’s financial support: This material is based upon work for the NSF under Award No. DMS-1309236.

#### Note

Second and third author’s financial support: Research partially supported by the MINECO/FEDER grant no. MTM2014-52368-P.

#### Article information

Source
J. Differential Geom., Volume 109, Number 3 (2018), 509-565.

Dates
First available in Project Euclid: 10 July 2018

https://projecteuclid.org/euclid.jdg/1531188195

Digital Object Identifier
doi:10.4310/jdg/1531188195

Mathematical Reviews number (MathSciNet)
MR3825610

Zentralblatt MATH identifier
06917316

#### Citation

Meeks, William H.; Pérez, Joaquín; Ros, Antonio. The local picture theorem on the scale of topology. J. Differential Geom. 109 (2018), no. 3, 509--565. doi:10.4310/jdg/1531188195. https://projecteuclid.org/euclid.jdg/1531188195