On complete minimal surfaces with finite Morse index in three manifolds (Q1063268)

The author obtains several good results on the Morse index (for area) of an oriented complete minimal surface (M,g) in a Riemannian 3-manifold N. Amongst them: 1) If index M\(<\infty\), there is a compact \(C\subset M\) so that \(M\setminus C\) is stable and there is a positive function \(u: M\to {\mathbb{R}}\) with \(Lu=0\) on \(M\setminus C\), where L is the second variation operator. If N has scalar curvature \(\geq 0\), then \(u^ 2g\) is a complete metric on M with Gaussian curvature \(\geq 0\) on \(M\setminus C\). In particular, M is conformally equivalent to a Riemann surface with a finite number of punctures. 2) If \(N={\mathbb{R}}^ 3\), then index M\(<\infty\) iff M has finite total curvature.