The Ricci flow on four-manifolds and the Seiberg-Witten equations

A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, R.Hamilton introduced this notion as a nice class of solutions and classified 3-dimensional non-singular solutions. In this talk, we shall consider non-singular solutions in dimension 4. By using gauge theoretical invariants associated with the Seiberg-Witten equations, we shall prove obstructions to the existence of non-singular solutions. One of the main ingredients of the proof is curvature bounds derived from the Seiberg-Witten equations.If time allows, we shall also discuss obstructions coming from a twisted version of the Seiberg-Witten equations and related problems.

海报