ABSTRACT
At certain times, a Ricci flow may develop singularity at some parts of the manifold but stay smooth in other parts. In order to prolong the Ricci flow beyond this time, one needs to cut off certain regions with high curvature, which are replaced by cap shaped manifold. The resulting manifold serves as the initial data for a new Ricci flow. This process is called a surgery. Due to the canonical neighborhood Theorem 7.5.1, for 3 dimensional Ricci flows, regions with higher curvature have simple topological and geometrical structures. The most essential ones are ǫ horns where surgeries take place. During a surgery, the singular part of an ǫ horn is cut off along a central 2 sphere, which is then pasted to a surgery cap that is diffeomorphic to the Euclidean three ball.
