On the topology and index of minimal surfaces

TL;DR

This paper establishes bounds on the index of two-sided minimal surfaces in R^3 based on genus, ends, and total curvature, proving nonexistence of certain minimal surfaces with low index.

Contribution

It provides new bounds linking the index of minimal surfaces to their topological and geometric properties, confirming a conjecture by Choe.

Findings

Lower bounds on index depending on genus and ends

Nonexistence of embedded minimal surfaces in R^3 with index 2

Index bounds proportional to total curvature

Abstract

We show that for an immersed two-sided minimal surface in $R^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $R^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of a immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.