  We generalize the construction given in math.AG/0309435 of a "constant"
t-structure on the bounded derived category of coherent sheaves $D(X\times S)$
starting with a t-structure on $D(X)$. Namely, we remove smoothness and
quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that
are not necessarily Noetherian but are close to Noetherian in the appropriate
sense. The main new tool is the construction of induced t-structures that uses
unbounded derived categories of quasicoherent sheaves and relies on the results
of \cite{AJS}. As an application of the "constant" t-structures techniques we
prove that every bounded nondegenerate t-structure on $D(X)$ with Noetherian
heart is invariant under the action of a connected group of autoequivalences of
$D(X)$. Also, we show that if $X$ is smooth then the only local t-structures on
$D(X)$, i.e., those for which there exist compatible t-structures on $D(U)$ for
all open $U\subset X$, are the perverse t-structures considered in
math.AG/0005152.
