Perelman has proved that there cannot exist a nontrivial breather for Ricci flow on closed manifold. Here we construct expanding breathers in all dimensions when the underlying manifold is allowed to be noncompact.

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S1 × S2, representing the canonical ensemble for gravity in a box. At high temperature the action has three ...

We estimate the diameter of a closed manifold evolving under Ricci flow in terms of a scalar curvature integral. The proof uses a new maximal function and extends some of Perelman’s recent ideas.