On complete minimal surfaces with finite Morse index in three manifolds
type: Item
instance of: scholarly article
title: On complete minimal surfaces with finite Morse index in three manifolds
zbMATH Open document ID: 0573.53038
DOI: 10.1007/BF01394782
author: D. Fischer-Colbrie
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143280
review text: 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 3915164
zbMATH Keywords: Morse index
zbMATH Keywords: minimal surface
zbMATH Keywords: second variation operator
zbMATH Keywords: total curvature
reviewed by: J. Eells
MaRDI profile type: Publication
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Q5611582
cites work: The Plateau problem for non-relative minima
description: scientific article; zbMATH DE number 3915164