2025/12/25 by Haiping Fu, Yao Lu, Fu, Haiping +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.2512.21496
openalex publication_date 2025/12/25 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28
Using Bochner techniques, we prove that a compact Einstein manifold of dimension n ≥ 4 has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity. Furthermore, employing a result of Li \citeLi5, we establish that any closed Einstein manifold of dimension n ≥ 4 satisfying k-1(λ1+⋯ +λk)≥ -θ(n,k) λ, for some k ≤ [(n+2)/(4)] must be either flat or a spherical space form. Here, λ1≤ λ2≤ ⋯ ≤ λ((n-1)(n+2))/(2) are the eigenvalues of \mathringR , λ is their average, and θ(n,k) is a positive constant. This result generalizes the work of Dai-Fu \citeDF and Chen-Wang \citeCW1,CW.We also classify four-dimensional Einstein manifolds satisfying a cone condition.