  We investigate the tautness of invariant Fatou components for holomorphic
endomorphisms of P^2. Previously, only basins of attraction were known to be
taut. We show that two other kinds of recurrent Fatou components are taut. In
the first of these cases, as well as for basins of attraction, we show that the
Fatou components are in fact Kobayashi complete, which implies tautness.
