Morse-type indices of of two-dimensional minimal surfaces in $\mathbf R^3$ and $\mathbf H^3$

A. A. Tuzhilin

Izvestiya. Mathematics , Tome 38 (1992) no. 3, pp. 575-598.

Résumé

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 $\mathbf R^3$ and in three-dimensional Lobachevsky space $\mathbf H^3$ are computed. In particular, the indices of all the classic minimal surfaces in $\mathbf R^3$ are computed: the catenoid, Enneper surfaces, Scherk surfaces, Richmond surfaces, and others. The indices of spherical catenoids in $\mathbf H^3$ are computed, which completes the computation of the indices of catenoids in $\mathbf H^3$ (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 $\mathbf H^3$ the helicoids are stable for certain values of the parameter.

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