On the topology and index of minimal surfaces II

TL;DR

This paper establishes a lower bound on the Morse index of immersed minimal surfaces in three-dimensional space based on their genus and ends, resolving several conjectures about the classification of low-index minimal surfaces.

Contribution

It provides improved estimates linking Morse index with genus and ends, and proves non-existence results for certain low-index minimal surfaces, advancing classification efforts.

Findings

No complete two-sided minimal surfaces in $ eal^3$ of index two.

No complete embedded minimal surfaces with index three.

No complete one-sided minimal immersions with index one.

Abstract

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there are no complete two-sided immersed minimal surfaces in $\mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.