Tohoku Mathematical Journal
2019
December
SECOND SERIES VOL. 71, NO. 4

Tohoku Math. J.
71 (2019), 581-607

Title $L^2$ CURVATURE PINCHING THEOREMS AND VANISHING THEOREMS ON COMPLETE RIEMANNIAN MANIFOLDS

Author Yuxin Dong, Hezi Lin and Shihshu Walter Wei

(Received November 17, 2017, revised January 4, 2018)
Abstract. 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.

Mathematics Subject Classification. Primary 53C20; Secondary 53C21, 53C25.
Key words and phrases. conformally flat, vanishing theorems, $L^2$ harmonic $p$-forms, ends, Liouville theorems.

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