Bounds on the topology and index of minimal surfaces

TL;DR

This paper establishes bounds on the number of ends and the stability index of complete, embedded minimal surfaces in three-dimensional space, based on their genus and finite topology.

Contribution

It proves that for each genus, there is a bound on the number of ends and the stability index of such minimal surfaces, linking topology to geometric stability.

Findings

Bound on the number of ends depends only on genus.

Stability index is finite and bounded by a genus-dependent constant.

Minimal surfaces with at least two ends have finite stability index.

Abstract

We prove that for every nonnegative 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.