Complete properly embedded minimal surfaces in $\mathbf{R}^3$

Abstract

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface $Σ^{2} ⊂\mathbf{R}^{3}$ with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature.