Spectral asymmetry and index theory on manifolds with generalised hyperbolic cusps

Abstract

We consider a complete Riemannian manifold, which consists of a compact interior and one or more ϕ-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here ϕ is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on ϕ. These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In these two special cases, and in an intermediate case, we give explicit expressions for the contribution from infinity in the index formula. Spectral asymmetry plays an important role, as it does in the special case of cylindrical ends studied by Atiyah-Patodi-Singer in the 1970s.