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# 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

#### 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.

Received 9 May 2016

Accepted 3 September 2019

Published 30 September 2019