contribute. On the other hand, one expects that the zero mode in the fiber (the space of fiber harmonic forms) makes a simlar contribution in both the cusp and cone cases, in particular that an incomplete cusp edge space which is not Witt will have an infinite dimensional space of closed extensions on which ‘Cheeger ideal boundary conditions’ must be imposed to make the operator self-adjoint, as is the case in [@ALMPII].

A second closely related work is Grieser-Hunsicker [@Grieser-Hunsicker], which uses also inhomogeneous radial blowups, in this case to construct a Green’s function for elliptic operators on a certain class of complete Riemannian manifolds (called ‘$\phi$-manifolds’) which require similar analysis. There are many other related works in a similar vein including, just to name a few, Albin-Rochon [@Albin-Rochon], Brüning-Seeley [@Brunin-Seeley], Gil-Krainer-Mendoza [@Gil-Krainer-Mendoza], Lesch [@L1997], Schultze [@Schulze], and Grieser’s notes on parametrix constructions for heat kernels [@Grieser-notes]. For analysis of moduli space, to give just a sample recent work, we refer the reader to the papers of Liu-Sun-Yao, for example [@Liu-Sun-Yau-goodness; @Liu-Sun-Yao-New-results].

Incomplete cusp edge differential geometry {#sec:differential-geometry}
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As described above we work on a smooth manifold with boundary $M$ whose boundary is the total space of a fiber 