Rigidity of five-dimensional quasi-Einstein manifolds with constant scalar curvature
Zhongxian Cao
TL;DR
The paper proves a rigidity result for nontrivial, simply-connected, compact 5-dimensional $m$-quasi-Einstein manifolds with boundary under constant scalar curvature. By employing the $L_{m+2}$ operator, detailed level-set geometry, and Gauss-Bonnet-Chern arguments on 4D level sets, the authors derive decay estimates for the trace-free Ricci part and control Weyl tensor contributions. They show that the two smallest eigenvalues of the trace-free Ricci part vanish outside a compact set, which, together with analyticity, forces the entire manifold to be rigid and split as a product up to scaling. The main result is that $M$ is isometric to the doubly warped product $\mathbb{S}_{+}^2\times\mathbb{S}^3$ with scalar curvature $R=\frac{3m+5}{m+1}\lambda$, thereby completing the $k=3$ rigidity classification in dimension five for these quasi-Einstein manifolds.
Abstract
Let $(M^5,g)$ be a five-dimensional non-trivial simply-connected compact quasi-Einstein manifold with boundary. If $M$ has constant scalar $R$, Johnatan Costa, Ernani Ribeiro Jr, and Detang Zhou show that $R$ = $((m-5)k+20)/(m-k+4)λ$ for some $k\in\{0,2,3,4\}$. Both cases of $k=0$ and $k=4$ are already classified. In this paper we will prove that the case $k=3$ is rigid.