The Yamabe invariants of orbifolds and cylindrical manifolds, and $L^2$-harmonic spinors

Abstract

We study the Yamabe invariants of cylindrical manifolds and compact orbifolds with a finite number of singularities, by means of conformal geometry and the Atiyah-Patodi-Singer $L^2$-index theory. For an $n$-orbifold $M$ with singularities $Σ_Γ = \{(\check{p}_1, Γ_1), ..., (\check{p}_s, Γ_s)\}$ (where each group $Γ_j<O(n)$ is of finite order), we define and study the \emph{orbifold Yamabe invariant} $Y^{\orb}(M)$. We prove that $Y^{\orb}(M)$ coincides with the corresponding $h$-$\emph{cylindrical Yamabe invariant}$ $Y^{h\textrm{-}\cyl}(M \setminus \{\check{p}_1, ..., \check{p}_s\})$ defined by the authors \cite{AB2}, where $h = h_{Γ_j}$ is the standard metric on the slice $S^{n-1}/Γ_j$ of each end with infinity $\check{p}_j$. Using this, we show that $Y^{\orb}(M)$ is bounded by $Y(S^n) /d$ from above, where $d=\max_j|Γ_j|^{\frac{2}{n}}$. For a cylindrical 4-manifold $X$ with a general slice metric $h$ on the end, we also establish a method for estimating the $h$-cylindrical Yamabe invariant $Y^{h\textrm{-}\cyl}(X)$ from above, in terms of the geometry and topology of $X$. We conclude by an explicit estimate of $Y^{h\textrm{-}\cyl}(X)$ for particular cylindrical 4-manifolds $X$, including that of $Y^{\orb}(M)$ for 4-orbifolds $M$.