In this talk, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killing spinors are strictly stable. Moreover, we show that Ricci-flat and hyperbolic cones over Kähler-Einstein Fano manifolds and over nonnegatively curved Einstein manifolds are stable if the cone has dimension n>=10.

Alberti proved that if A is a m-dimensional closed set in $\mathbf{R}^{m}$, if k=1, ..., m-1 and $\Sigma^{k}(A) = \{ x \in \partial\,A : \mathscr{H}^{m-k}(\mathscr{N}(A,x) > 0 \}$ then $\Sigma^{k}(A)$ is countably $(\mathscr{H}^{k}.k)$ rectifiable of class 2.   In this talk we introduce a new notion of unit normal bundle that it allows to suitably generalize the Alberti theorem to any closed set.   Assuming a Lusin type property for the unit normal bundle we study the second order rectifiability properties of k-dimensional closed sets in $\mathbf{R}^{m}$.