2023/08/05 by Xu, Yiyan, Zhang, Shihong
#53C18 #53C24 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.02777
In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented n-dimensional (n≥6) Riemannian manifold (M,g) and prove the following results under the condition ∫M ∇ R⋅∇ Qd Vg≤0. (1) If (M,g) is locally conformally flat with nonnegative Ricci curvature, then (M,g) is isometric to a quotient of ℝn, \mathbbSn, or ℝ×\mathbbSn-1. (2) If (M,g) has δ2 W=0 with nonnegative sectional curvature, then (M,g) is isometric to a quotient of the product of Einstein manifolds. Additionally, we investigate some rigidity theorems involving Q-curvature about hypersurfaces in simply-connected space forms. We also show the uniqueness of metrics with constant scalar curvature and constant Q-curvature in a fixed conformal class.