Abstract
Let $X$ be a compact manifold with boundary $\partial X$, and suppose that $\partial X$ is the total space of a fibration \[ Z\rightarrow \partial X \rightarrow Y\, . \] Let $D_\Phi$ be a generalized Dirac operator associated to a $\Phi$-metric $g_\Phi$ on $X$. Under the assumption that $D_\Phi$ is fully elliptic we prove an index formula for $D_\Phi$. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain $b$-metric $g_b (\epsilon)$. Next we write the $b-$ (i.e. the APS) index formula for $g_b(\ep)$; the $\Phi$-index formula follows by analyzing the limiting behaviour as $\epsilon\searrow 0$ of the two terms in the formula. The interior term is studied directly whereas the adiabatic limit formula for the eta invariant follows from work of Bismut and Cheeger.
Citation
Eric Leichtnam. Rafe Mazzeo. Paolo Piazza. "The index of Dirac operators on manifolds with fibered boundaries." Bull. Belg. Math. Soc. Simon Stevin 13 (5) 845 - 855, January 2007. https://doi.org/10.36045/bbms/1170347808
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