On the topology and index of minimal surfaces

Abstract

We show that for an immersed two-sided minimal surface in $\mathbb{R}^3$, 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 $\mathbb{R}^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of an 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.