  We provide in this work a semigroup approach to the study of singular PDEs,
in the line of the paracontrolled approach developed recently by Gubinelli,
Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional
calculus and introduce a paraproduct based on the semigroup, for which
commutator estimates and Schauder estimates are proved, together with their
paracontrolled extensions. This machinery allows us to investigate singular
PDEs in potentially unbounded Riemannian manifolds under mild geometric
conditions. As an illustration, we study the generalized parabolic Anderson
model equation and prove, under mild geometric conditions, its well-posed
character in Holders spaces, in small time on a potentially unbounded
2-dimensional Riemannian manifold, for an equation driven by a weighted noise,
and for all times for the linear parabolic Anderson model equation in
2-dimensional unbounded manifolds. This machinery can be extended to an even
more singular setting and deal with Sobolev scales of spaces rather than Holder
spaces.
