$L^2$ curvature pinching theorems and vanishing theorems on complete Riemannian manifolds

TL;DR

This paper establishes new $L^2$ curvature pinching and vanishing theorems for complete Riemannian manifolds, particularly focusing on locally conformally flat manifolds with scalar curvature conditions, and explores implications for harmonic and $p$-harmonic functions.

Contribution

It introduces general $L^2$ rigidity theorems and vanishing results for harmonic forms on LCF manifolds under curvature pinching and integral conditions, extending previous understanding.

Findings

Rigidity theorems for LCF manifolds with constant scalar curvature.

Vanishing results for harmonic $p$-forms under curvature conditions.

Liouville theorems for $p$-harmonic functions on LCF manifolds.

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.