Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusions

It is well known that the classical Sobolev inequality not only holds on Euclidean space, but also on any Cartan-Hadamard manifold, namely a complete and simply connected Riemannian manifold with everywhere nonpositive sectional curvatures. On the other hand, the Poincaré (or spectral gap) inequality fails on Euclidean space but holds on hyperbolic space or more in general on any Cartan-Hadamard manifold with sectional curvatures bounded from above by a negative constant. However, almost nothing seems to be known in between, that is when curvatures are negative but allowed to vanish at infinity. Here we show some partial results in this direction, mainly restricted to radial functions, and discuss related consequences concerning smoothing effects for certain nonlinear diffusions of porous medium type. This is a joint work with A. Roncoroni.