L2 CURVATURE PINCHING THEOREMS AND VANISHING THEOREMS ON COMPLETE RIEMANNIAN MANIFOLDS

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.