MORSE-TYPE INDICES OF TWO-DIMENSIONAL MINIMAL SURFACES IN R3 AND H3

Abstract

The Morse-type index of a compact p-dimensional minimal submanifold is the index of the second variation of the p-dimensional volume functional. In this paper a definition is given for the index of a noncompact minimal submanifold, and the indices of some two-dimensional minimal surfaces in three-dimensional Euclidean space R3 and in three-dimensional Lobachevsky space H3 are computed. In particular, the indices of all the classic minimal surfaces in R3 are computed: the catenoid, Enneper surfaces, Scherk surfaces, Richmond surfaces, and others. The indices of spherical catenoids in H3 are computed, which completes the computation of the indices of catenoids in H3 (hyperbolic and parabolic catenoids have zero index, that is, they are stable). It is also proved that for a one-parameter family of helicoids in H3 the helicoids are stable for certain values of the parameter.