We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the 3-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

This erratum corrects an error arising in the proof of Proposition 6.4 in the article “On the density of geometrically finite Kleinian groups”

We consider a 1-parameter family of strictly convex hypersurfaces in Rn+1 moving with speed −Kαν, where ν denotes the outward-pointing unit normal vector and α⩾1/(n+2). For α>1/(n+2), we show that the flow converges to a round sphere after rescaling. In the affine invariant case α=1/(n+2), our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.