On complete minimal surfaces with finite Morse index in three manifolds
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.
scientific article; zbMATH DE number 3915164