Regularity of the eta function on manifolds with cusps

TL;DR

This paper investigates the eta function of the Dirac operator on manifolds with cusps, proving regularity properties and entireness under certain geometric conditions, which advances understanding of spectral invariants in non-compact settings.

Contribution

It establishes conditions under which the eta function is regular or entire on manifolds with cusps, extending spectral analysis to non-compact hyperbolic manifolds.

Findings

Eta function has at most simple poles under certain conditions.

Eta function is entire for hyperbolic manifolds of finite volume.

Regularity results depend on invertibility at infinity.

Abstract

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire.