Communications in Analysis and Geometry

Volume 29 (2021)

Number 2

An APS index theorem for even-dimensional manifolds with non-compact boundary

Pages: 293 – 327



Maxim Braverman (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)

Pengshuai Shi (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)


We study the index of the APS boundary value problem for a strongly Callias-type operator $\mathcal{D}$ on a complete Riemannian manifold $M$. We use this index to define the relative $\eta$-invariant $\eta (\mathcal{A}_1 , \mathcal{A}_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the $\eta$-invariants of $\mathcal{A}_1$ and $\mathcal{A}_0$ are not defined, the relative $\eta$-invariant behaves as if it were the difference $\eta (\mathcal{A}_1) - \eta (\mathcal{A}_0)$. We also define the spectral flow of a family of such operators and use it to compute the variation of the relative $\eta$-invariant.

Maxim Braverman was partially supported by the Simons Foundation collaboration grant #G00005104.

Received 6 March 2018

Accepted 26 November 2018

Published 19 April 2021