Degenerate neckpinches in Ricci flow
  In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer $k\geq3$, there exist compact solutions in all dimensions $m\geq3$ that become singular at the rate (T-t)^{-2+2/k}$. 
Sigurd B. Angenent, James Isenberg, and Dan Knopf
math.DG gr-qc