Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds

2025/12/21 by Yunhee Euh, Euh, Yunhee, JeongHyeong Park +1
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2512.18758

Abstract

We study critical metrics of the curvature functional \A(g)=∫_M |R|² \vol, on complete four-dimensional Riemannian manifolds (M,g) with finite energy, that is, \A(g)<∞. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that (M,g) is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures c and -c (c≠ 0). This extends the compact classification of four-dimensional A-critical metrics obtained in earlier work to the complete setting.

Citations

• Einstein manifolds and curvature operator of the second kind
• Curvature of the Second kind and a conjecture of Nishikawa
• Rigidity of Critical Metrics for Quadratic Curvature Functionals
• On Variational Properties of Quadratic Curvature Functionals
• Critical Metrics for Riemannian Curvature Functionals
• A Curvature identity on a 4-dimensional Riemannian manifold
• Extrema of curvature functionals on the space of metrics on 3-manifolds, II

Related

• Some rigidity characterizations on critical metrics for quadratic curvature functionals
• Uniqueness of critical metrics for a quadratic curvature functional
• Four-dimensional homogeneous critical metrics for quadratic curvature functionals
• Critical metrics of the total scalar curvature functional on 4-manifolds
• Rigidity of Critical Metrics for Quadratic Curvature Functionals