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2019 $L^2$ curvature pinching theorems and vanishing theorems on complete Riemannian manifolds
Yuxin Dong, Hezi Lin, Shihshu Walter Wei
Tohoku Math. J. (2) 71(4): 581-607 (2019). DOI: 10.2748/tmj/1576724795

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.

Citation

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Yuxin Dong. Hezi Lin. Shihshu Walter Wei. "$L^2$ curvature pinching theorems and vanishing theorems on complete Riemannian manifolds." Tohoku Math. J. (2) 71 (4) 581 - 607, 2019. https://doi.org/10.2748/tmj/1576724795

Information

Published: 2019
First available in Project Euclid: 19 December 2019

zbMATH: 07199981
MathSciNet: MR4043927
Digital Object Identifier: 10.2748/tmj/1576724795

Subjects:
Primary: 53C20
Secondary: 53C21, 53C25

Rights: Copyright © 2019 Tohoku University

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Vol.71 • No. 4 • 2019
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