A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

2010/05/25 by Jörn Müller, Müller, Jörn
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1005.4606

openalex publication_date 2010/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of ℚ-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology H^p(X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge-Laplace-Operator on Ω^p(X).

Citations

• Index and Spectral Theory for Manifolds with Generalized Fibred Cusps

Related

• Extended Hodge Theory for Fibred Cusp Manifolds
• Resolvent, heat kernel and torsion under degeneration to fibered cusps
• Families index for manifolds with hyperbolic cusp singularities
• Lefschetz theorems, Hodge-Riemann relations and Ample vector bundles
• Hodge cohomology of iterated fibred cusp metrics on Witt spaces