2021/10/06 by Giovanni Catino, Catino, Giovanni, Paolo Mastrolia +3 · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2110.02683
openalex publication_date 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals \mathfrakF2t = ∫ |Ricg|2 dVg + t ∫ R2g dVg, t∈ℝ, and \mathfrakS2 = ∫ Rg2 dVg. We show that (i) flat surfaces are the only critical points of \mathfrakS2, (ii) flat three-dimensional manifolds are the only critical points of \mathfrakF2t for every t>-(1)/(3), (iii) three-dimensional scalar flat manifolds are the only critical points of \mathfrakS2 with finite energy and (iv) n-dimensional, n>4, scalar flat manifolds are the only critical points of \mathfrakS2 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M. T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.