  We establish continuous maximal regularity results for parabolic differential
operators acting on sections of tensor bundles on Riemannian manifolds. As an
application, we show that solutions to the Yamabe flow instantaneously
regularize and become real analytic in space and time. The regularity result is
obtained by introducing a family of parameter-dependent diffeomorphims acting
on functions over Riemannian manifolds in conjunction with maximal regularity
and the implicit function theorem.
