Index, Vision Number and Stability of Complete Minimal Surfaces JAIGYOUNG CHOE

Let Z be a complete minimal surface in R a. Z is said to be stable if2: minimizes area up to second order on each compact set. If K is the Gauss curvature, then the condition that Z be stable is expressed analytically by the requirement that on any relatively compact domain D of Z, the first eigenvalue of the Jacobi operator L---- A-- 2K be positive. Here L is defined in the space Fo(N(Z)) of all smooth compactly supported normal vector fields on Z vanishing on OD. For a relatively compact domain D of Z, the index of D is defined by the number of negative eigenvalues of the operator A-- 2K. Z is said to have finite index if the index of every relatively compact domain has a uniform upper bound. Denote the least upper bound by Ind (Z). If2J is stable then Ind (Z) = 0 and vice versa. Recently FISCHER-COLBRIE [17] showed that oriented Z has finite index if and only if 2J has finite total curvature. If Z has finite total curvature, it was shown by OSSER-MAN [O] that Z is conformally a compact Riemann surface Ewith finite punctures. In this paper we compute a lower bound for Ind (Z) in terms of a new geome-tric quantity, the vision number of Z. The key idea is to apply the generalized Morse index theorem of S. SMALE [Sin] to a bounded Jacobi field on Z constructed from a suitable Killing vector field in the ambient space. As a result, we show that i) the index of JORGE & MEEKS ' minimal surface with k ends [JM] is at leas