Open Access
July 2018 The local picture theorem on the scale of topology
William H. Meeks, Joaquín Pérez, Antonio Ros
Author Affiliations +
J. Differential Geom. 109(3): 509-565 (July 2018). DOI: 10.4310/jdg/1531188195


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.

Funding Statement

First author’s financial support: This material is based upon work for the NSF under Award No. DMS-1309236.
Second and third author’s financial support: Research partially supported by the MINECO/FEDER grant no. MTM2014-52368-P.


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William H. Meeks. Joaquín Pérez. Antonio Ros. "The local picture theorem on the scale of topology." J. Differential Geom. 109 (3) 509 - 565, July 2018.


Received: 9 February 2016; Published: July 2018
First available in Project Euclid: 10 July 2018

zbMATH: 06917316
MathSciNet: MR3825610
Digital Object Identifier: 10.4310/jdg/1531188195

Primary: 53A10
Secondary: 49Q05 , 53C42

Keywords: curvature estimates , finite total curvature , injectivity radius , limit tangent cone , locally simply connected , minimal lamination , minimal parking garage structure , minimal surface , removable singularity , stability

Rights: Copyright © 2018 Lehigh University

Vol.109 • No. 3 • July 2018
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