  This work examines the critical anisotropy required for the local stability
of the collinear ground states of a geometrically-frustrated triangular-lattice
antiferromagnet (TLA). Using a Holstein-Primakoff expansion, we calculate the
spin-wave frequencies for the 1, 2, 3, 4, and 8-sublattice (SL) ground states
of a TLA with up to third neighbor interactions. Local stability requires that
all spin-wave frequencies are real and positive. The 2, 4, and 8-SL phases
break up into several regions where the critical anisotropy is a different
function of the exchange parameters. We find that the critical anisotropy is a
continuous function everywhere except across the 2-SL/3-SL and 3-SL/4-SL phase
boundaries, where the 3-SL phase has the higher critical anisotropy.
