The Seiberg–Witten equations on end‐periodic manifolds and an obstruction to positive scalar curvature metrics

2016/03/31 by Jianfeng Lin · 4 citations
Mathematics · #Advanced Operator Algebra Research #Curvature #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Pure mathematics #Scalar (mathematics) #Scalar curvature #math.GT

paper · pdf · doi:10.1112/topo.12090

published in Journal of Topology 12(2), 328-371 (Wiley) · v3: Added Section 4.2 for a detailed proof of exponential decay

arxiv created 2019/01/13 · openalex publication_date 2019/01/25 · arxiv updated 2019/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

By studying the Seiberg–Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S 1 × S 3 . This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of H 3 ( X ; Z ) with the 4-dimensional Casson invariant λ S W ( X ) defined in [Mrowka, Ruberman and Saveliev, J. Differential Geom. 88 (2011) 333–377]. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.

Citations

• Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
• Gauge theory on asymptotically periodic 4-manifolds
• Dirac operators on manifolds with periodic ends
• On the structure of manifolds with positive scalar curvature
• Monopole Floer homology for rational homology 3-spheres
• Seiberg–Witten–Floer stable homotopy type of three-manifolds withb1= 0