A Kummer construction for gravitational instantons

We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.Comment: The construction of locally hyperkahler ALH spaces is corrected and complete

Extinction in a finite time for parabolic equations of fast diffusion type on manifolds

We prove extinction in a finite time for a singular parabolic equation on a Riemannian manifold, under suitable assumptions on the Riemannian metric and on the inhomogeneous coefficient appearing in the equation. The result relies on a suitable embedding theorem, of which we present a new proof