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Morse-type indices of of two-dimensional minimal surfaces in $\mathbf R^3$ and $\mathbf H^3$
A. A. Tuzhilin
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
$\mathbf R^3$ 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.
UDC:
514.77
MSC: Primary
53A10,
49Q05; Secondary
53C42 Received: 22.10.1987
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