A slice theorem for open manifolds (Q1611625)

For an open Riemannian manifold \((M^n,g_0)\) with positive injectivity radius and whose curvature tensor and all its covariant derivatives up to order \(k\) are uniformly bounded let \(\mathcal M\) be the space of metrics. Let \(M^n\) be of strictly negative curvature \(K_{g_0}\leq -c<0\) with \(\inf \sigma_e(\Delta_0(g_0))>0\). For \(r>n/2+2\) let \(\text{comp}^r(g_0)_{_{K <0}}\) be the component submanifold in the completed space \(\mathcal M^r\) of metrics which consist of metrics of bounded geometry and have strictly negative sectional curvature. The group \(\mathcal D^{r+1}\) of diffeomorphisms of the manifold \(M^n\) consists of diffeomorphisms \(f\) with bounded \(r\)-Sobolev norm. The group \(\mathcal D^{r+1}_0\) is the identity component of the completed diffeomorphism group \(\mathcal D^{r+1}\). In this paper, the author establishes a slice theorem for the components \(\text{comp}^r(g_0)_{_{K <0}}\) and proves that \(\mathcal D^{r+1}_0\) acts properly on \(\text{comp}^r(g_0)_{_{K <0}}\) and \(g\cdot \mathcal D^{r+1}_0\) admits a slice. The author applies this theorem to the Teichmüller theory for open surfaces.