  In this paper, by using monotonicity formulas for vector bundle-valued
$p$-forms satisfying the conservation law, we first obtain general $L^2$ global
rigidity theorems for locally conformally flat (LCF) manifolds with constant
scalar curvature, under curvature pinching conditions. Secondly, we prove
vanishing results for $L^2$ and some non-$L^2$ harmonic $p$-forms on LCF
manifolds, by assuming that the underlying manifolds satisfy pointwise or
integral curvature conditions. Moreover, by a Theorem of Li-Tam for harmonic
functions, we show that the underlying manifold must have only one end.
Finally, we obtain Liouville theorems for $p$-harmonic functions on LCF
manifolds under pointwise Ricci curvature conditions.
