# 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, Richard Hamilton introduced it as an important special class of solutions and classied 3-dimensional non-singular solutions.In particular,it was proved that the underlying 3-manifold is geometrizable in the sense of Thurston. On the other hand, in 1994, new invariants of smooth 4-manifolds were introduced by Edward Witten. The invariants are constructed from nonlinear partial differential equations which are called the Seiberg-Witten equations. In this talk,the Seiberg-Witten equations are used to study the properties of 4-dimensional non-singular solutions.In particular,we would like to explain that gauge theoretical invariants associated with the Seiberg-Witten equations give rise to obstructions to the existence of 4-dimensional non-singular solutions and we will also discuss its application.