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On complete minimal surfaces with finite Morse index in three manifolds. (English) Zbl 0573.53038
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.
Reviewer: J.Eells

MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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References:
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