On some rigidity theorems of Q-curvature

Abstract

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\geq6$) Riemannian manifold $(M,g)$ and prove the following results under the condition $\int_{M} \nabla R\cdot\nabla \mathrm{Q}\mathrm{d} V_g\leq0$. (1) If $(M,g)$ is locally conformally flat with nonnegative Ricci curvature, then $(M,g)$ is isometric to a quotient of $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{R}\times\mathbb{S}^{n-1}$. (2) If $(M,g)$ has $\delta^2 W=0$ 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.