Journal of Differential Geometry

The local picture theorem on the scale of topology

William H. Meeks, Joaquín Pérez, and Antonio Ros

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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.


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.

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

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


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.

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