Index, vision number and stability of complete minimal surfaces (Q910025)

A new geometric quantity, the vision number, is defined. By using this vision number combined with the Morse-Smale index theorem the author finds lower bounds for the index of minimal surfaces of finite total curvature, mainly, in \(R^ 3\) and in \(S^ 3\). He computes lower bounds for the index of the classical genus zero surfaces of Jorge-Meeks in \(R^ 3\), of the three-punctured genus \(g\geq 1\) embedded minimal surfaces of Costa, Hoffman and Meeks in \(R^ 3\) and of the Lawson's embedded minimal surfaces in \(S^ 3\). Also, some results in global stability of non-orientable minimal surfaces are derived. For example, that any complete minimal surface in \(R^ 3\) of total curvature -2\(\pi\) is stable, and that a complete minimal surface in \(R^ 3\) with finite total curvature conformally equivalent to a Klein bottle punctured at a finite number of points is unstable.