Ricerc@Sapienza

We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide.

We give an explicit description of the non-flat parallel even Clifford structures of rank 8, 6, 5 on some real, complex and quaternionic Grassmannians, and discuss the rôle of the octonions in them, in particular for some low dimensional examples.

We build blowing-up solutions for linear perturbation of the Yamabe problem
on manifolds with boundary, provided the dimension of the manifold is n ≥ 7
and the trace-free part of the second fundamental form is non-zero everywhere on the
boundary

We built sign-changing solutions for a linear perturbation of the classical Yamabe problem.

We construct families of positive solutions for competitive and cooperative systems
which blow-up and concentrate at different points of the domain.
This problem can be seen as a generalization for systems of a Brezis–
Nirenberg type problem.