Metric stretching and the period map for smooth 4-manifolds

2023/09/28 by Christopher Scaduto, Scaduto, Christopher · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2309.16820

openalex publication_date 2023/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The period map for a smooth closed 4-manifold assigns to a Riemannian metric the space of self-dual harmonic 2-forms. This map is from the space of metrics to the Grassmannian of maximal positive subspaces in the second cohomology, where positivity is defined by cup product. We show that the period map has dense image for every 4-manifold, and that it is surjective if b⁺=1. Similar results hold for manifolds of dimension a multiple of four. The proofs involve families of metrics constructed by stretching along various hypersurfaces.

Cited by

• Anti-self-dual blowups II

Related

• Four-manifold systoles and surjectivity of period map
• 4-dimensional Riemannian manifolds with a harmonic 2-form of constant length
• A short proof of the surjectivity of the period map on K3 manifolds
• Geography of 4-manifolds with positive scalar curvature
• Harmonic 2-forms and positively curved 4-manifolds