The branching curves and their application to the four dimensional Ricci flow

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We study the four dimensional Ricci flow with the help of local invariants. If (M⁴, g(t)) is a solution to the Ricci flow and x ∈ M⁴, we can associate to the point x a one-parameter family of curves, which lie in the product of two projective lines. This allows us to reformulate the Cheeger-Gromov-Hamilton Compactness Theorem in the context of these curves. We use this result, in order to study Type I singularities in dimension four and give a characterization of the corresponding singularity models.