Acta Mathematica

Volume 223 (2019)

Number 1

Bounds on the topology and index of minimal surfaces

Pages: 113 – 149

DOI: https://dx.doi.org/10.4310/ACTA.2019.v223.n1.a2

Authors

William H. Meeks, III (Department of Mathematics, University of Massachusetts, Amherst, Mass., U.S.A.)

Joaquín Pérez (Department of Geometry and Topology, and Institute of Mathematics (IEMath-GR), University of Granada, Spain)

Antonio Ros (Department of Geometry and Topology, and Institute of Mathematics (IEMath-GR), University of Granada, Spain)

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.

Keywords

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

2010 Mathematics Subject Classification

Primary 53A10. Secondary 49Q05, 53C42.

Full Text (PDF format)

Received 9 May 2016

Accepted 3 September 2019

Published 30 September 2019