3 problems

Let $X$ be an $n$-dimensional Riemannian manifold. For each $q$ with $0\leq q\leq n$, let $\Delta^{(q)}$ be the Laplace operator on differential $q$-forms, let $h_t^{(q)}$ be its a…

Let $E$ be the regularized vacuum energy in dimension $3$, let $\Delta=(\kappa_1-\kappa_2)^2$, and let the ultraviolet regularization parameter tend to $0$. Compensating boundary t…

Let $M$ be a compact manifold, let … be a noncompact amenable Galois cover, and let $\{D_k\}_{k=1}^{\infty}$ be a regular exhaustion of $\widehat{M}$. Write $b^j(D_k,\partial D_k)$…