Index and flat ends of minimal surfaces (Q690131)

The purpose of this paper is to study the index of a complete orientable minimal surface in \(\mathbb{R}^3\) with finite total curvature. Theorem A. Let \(M\) be a generic complete orientable finitely branched minimal surface of genus zero in \(\mathbb{R}^3\) with finite total curvature \(4\pi d\). Then we have \[ \text{Index} (M)=2d-1\text{ and Nullity} (M)=3. \] Theorem B. A complete orientable finitely branched minimal surface in \(\mathbb{R}^3\) with finite total curvature has nullity \(\geq 4\) if and only if its Gauss map can be the Gauss map of a flat-ended minimal surface.