Geodesic
Morse-type indices of of two-dimensional minimal surfaces in $\mathbf R^3$ and $\mathbf H^3$
Izvestiya. Mathematics , Tome 38 (1992) no. 3, pp. 575-598

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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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     author = {A. A. Tuzhilin},
     title = {Morse-type indices of of two-dimensional minimal surfaces in $\mathbf R^3$ and $\mathbf H^3$},
     journal = {Izvestiya. Mathematics },
     pages = {575--598},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1992_38_3_a6/}
}
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A. A. Tuzhilin. Morse-type indices of of two-dimensional minimal surfaces in $\mathbf R^3$ and $\mathbf H^3$. Izvestiya. Mathematics , Tome 38 (1992) no. 3, pp. 575-598. http://geodesic.mathdoc.fr/item/IM2_1992_38_3_a6/