Acta Mathematica

Bounds on the topology and index of minimal surfaces

William H. Meeks, III, Joaquín Pérez, and Antonio Ros

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Abstract

We prove that for every non-negative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.

Article information

Source
Acta Math., Volume 223, Number 1 (2019), 113-149.

Dates
Received: 9 May 2016
First available in Project Euclid: 16 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.acta/1587002478

Digital Object Identifier
doi:10.4310/ACTA.2019.v223.n1.a2

Mathematical Reviews number (MathSciNet)
MR4018264

Zentralblatt MATH identifier
1428.53018

Subjects
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]

Keywords
minimal surface index of stability curvature estimates finite total curvature minimal lamination removable singularity

Citation

Meeks, William H.; Pérez, Joaquín; Ros, Antonio. Bounds on the topology and index of minimal surfaces. Acta Math. 223 (2019), no. 1, 113--149. doi:10.4310/ACTA.2019.v223.n1.a2. https://projecteuclid.org/euclid.acta/1587002478


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