Weyl energy and connected sums of four-manifolds

2025/05/23 by Malchiodi, Andrea, Malizia, Francesco
#49J99 #53C21 #58E11 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.17752

Abstract

Given two closed, oriented Riemannian four-manifolds (M,g_M) and (Z,g_Z), which are not locally conformally flat and not both self-dual or both anti-self-dual, we prove that there exists a metric g_Y on the connected sum Y≅ M#Z such that the Weyl energy of g_Y is strictly smaller than the sum of Weyl energies of g_M and g_Z.

Citations

• The Anti-Self-Dual Deformation Complex and a conjecture of Singer
• Curvature in the Balance: The Weyl Functional and Scalar Curvature of 4-Manifolds
• Optimal coordinates for Ricci-flat conifolds
• Integrability of Einstein deformations and desingularizations
• Integrability of Einstein deformations and desingularizations
• The renormalized volume of a 4-dimensional Ricci-flat ALE space
• Critical Metrics for Riemannian Curvature Functionals
• Weyl Curvature, Del Pezzo Surfaces, and Almost-Kaehler Geometry
• Critical metrics on connected sums of Einstein four-manifolds
• Desingularisation de metriques d'Einstein. II
• Short-time existence for some higher-order geometric flows
• Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat Metrics
• Canonical metrics on 3-manifolds and 4-manifolds
• On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth

Related

• Compact four-manifolds with pinched self-dual Weyl curvature
• Four dimensional closed manifolds admit a weak harmonic Weyl metric
• Pinched self-dual Weyl curvature on Einstein four-manifolds
• Certain 4-manifolds with non-negative sectional curvature
• On Optimal 4-Dimensional Metrics