Families index for manifolds with hyperbolic cusp singularities

TL;DR

This paper extends index theory for Dirac-type operators on manifolds with hyperbolic cusp singularities, broadening the class of boundary conditions and including families of operators, thus advancing geometric analysis in singular spaces.

Contribution

It generalizes existing index theorems to include families of perturbed Dirac operators on fibered hyperbolic cusp manifolds, relaxing boundary hypotheses.

Findings

Extended Vaillant's index treatment to broader boundary conditions.

Developed index theorem for families of perturbed Dirac operators.

Applied to manifolds with hyperbolic cusp singularities.

Abstract

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered boundary metrics).