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\(L^2\) curvature pinching theorems and vanishing theorems on complete Riemannian manifolds. (English) Zbl 1439.53036
Summary: 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.

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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