On some rigidity theorems of Q-curvature

TL;DR

This paper explores the rigidity properties of Q-curvature on higher-dimensional manifolds, establishing conditions under which the manifold's geometry is uniquely determined, and extends results to hypersurfaces and conformal classes.

Contribution

It proves new rigidity theorems for Q-curvature on closed manifolds with specific curvature conditions, including classifications of manifolds and hypersurfaces.

Findings

Manifolds with nonnegative Ricci curvature and locally conformally flat structure are isometric to standard models.

Manifolds with vanishing second divergence of the Weyl tensor and nonnegative sectional curvature are classified as quotients of Einstein manifolds.

Uniqueness of metrics with constant scalar and Q-curvature within a conformal class is established.

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.