Timothy Carson

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing previous examples. The class also includes metrics with non-compact singular ends which become instantaneously compact. Furthermore, we prove local stability of the forward evolution, which allows us to glue it to other manifolds and create a forward evolution from spaces which are not globally warped products.

We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow encountering a Type-I singularity modeled on ${\mathbb{R}}^{p+1}\times S^q$. This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from $D^{p+1}\times S^q$ to $S^p\times D^{q+1}$ (through the cone over $S^p\times S^q$.) We work in the class of doubly-warped product metrics. We also briefly discuss some possible smooth and non-smooth forward evolutions from other singular initial data.

Warped product metrics are a class of Riemannian metrics on cross products $B\times F$ which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base $B$ has two dimensions, the metric must in fact be a cross product. The theorem is a consequence of a Pohozaev-type identity. This closely parallels results in the study of blow-ups of solutions to the reaction- diffusion equation ${\partial }_{t}v=\Delta v+e^v$ in ${\mathbb{R}}^2$ .