  This is the first of a series of two papers which studies the fractional
porous medium equation on a Riemannian manifold with isolated conical
singularities. In this article, we show $R$-sectoriality for the fractional
powers of possibly non-invertible $R$-sectorial operators. Applications concern
existence, uniqueness and maximal $L^{q}$-regularity results for solutions of
the fractional porous medium equation on manifolds with conical singularities.
Space asymptotic behavior of the solutions close to the singularities is
provided and its relation to the local geometry is established. Our method
extends the freezing-of-coefficients method to the case of non-local operators
that are expressed as linear combinations of terms in the form of a product of
a function and a fractional power of a local operator.
