  We consider very weak solutions of the Cauchy problem for the porous medium
equation on Cartan-Hadamard manifolds, that are assumed to satisfy general
curvature bounds and to be stochastically complete. We identify a class of
initial data that can grow at infinity at a prescribed rate, which depends on
the assumed curvature bounds through an integral function, such that the
corresponding solution exists at least on $[0,T]$ for a suitable $T>0$. The
maximal existence time $T$ is estimated in terms of a suitable weighted norm of
the initial datum. Our results are sharp, in the sense that slower growth rates
yield global existence, whereas one can construct data with critical growth for
which the corresponding solutions blow up in finite time. Under further
assumptions, uniqueness of very weak solutions is also proved, in the same
growth class.
