Harmonic forms on manifolds with edges

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$

A heat trace anomaly on polygons

Let $\Omega_0$ be a polygon in \RR^2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that \Omega_\e is a family of surfaces with \calC^\infty boundary which converges to $\Omega_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as \e \searrow 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.Comment: Revision includes treatment of the Neumann problem and a discussion of the higher dimensional case; some new reference

Fredholm theory for elliptic operators on quasi-asymptotically conical spaces

We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such operators are Fredholm when between certain weighted Sobolev or weighted H\"older spaces. These are generalizations of well-known theorems in the asymptotically conical (or asymptotically Euclidean) setting, and also sharpen and extend corresponding theorems by Joyce. The methods here are based on heat kernel estimates originating from old ideas of Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As demonstrated by Joyce's work, the QAC spaces here contain many examples of gravitational instantons, and this work is motivated by various applications to manifolds with special holonomy