Heat-type equations on manifolds with fibered boundary I: Schauder estimates

In this paper we prove parabolic Schauder estimates for the Laplace-Beltrami operator on a manifold $M$ with fibered boundary and a $\Phi$-metric $g_\Phi$. This setting generalizes the asymptotically conical (scattering) spaces and includes special cases of magnetic and gravitational monopoles. This paper, combined with part II, lay the crucial groundwork for forthcoming discussions on geometric flows in this setting; especially the Yamabe- and mean curvature flow.