Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps

TL;DR

This paper develops an equivariant index theorem for Dirac operators on manifolds with generalized hyperbolic cusps, extending existing theories to include various cusp geometries and analyzing spectral symmetry conditions.

Contribution

It introduces a new index theorem for Dirac operators on manifolds with -cusps, encompassing cylindrical and hyperbolic ends, and relates cusp contributions to spectral symmetry.

Findings

Cusp contributions equal the delocalised -invariant in cylindrical cases

Index theorem reduces to Donnelly's theory for cylindrical ends

Cusp contribution vanishes when Dirac spectrum is symmetric

Abstract

We consider a complete Riemannian manifold, which consists of a compact interior and one or more $\varphi$-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here $\varphi$ 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 $\varphi$. These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised $\eta$-invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.