A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

TL;DR

This paper proves a Hodge-type theorem for manifolds with fibered cusp metrics, identifying harmonic representatives of de Rham cohomology using special values of eigenforms, extending results known for locally symmetric spaces.

Contribution

It establishes a Hodge-type theorem for fibered cusp manifolds, linking harmonic forms to eigenforms of the Hodge-Laplace operator, generalizing previous results.

Findings

Harmonic representatives of cohomology are given by special values of eigenforms.

The theorem extends Hodge theory to a new class of non-compact manifolds.

Connections to locally symmetric spaces are elucidated.

Abstract

A manifold with fibered cusp metrics $X$ can be considered as a geometrical generalization of locally symmetric spaces of $\mathbb{Q}$-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 $\Omega^p(X)$.