Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Authors: Hochs, Peter, Wang, Hang
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211920
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps. Peter Hochs and Hang Wang. SIGMA 19 (2023), 023, 32 pages

We consider a complete Riemannian manifold, which consists of a compact interior and one or more -cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here, 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 . These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised -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.