On the asymptotic scalar curvature ratio of complete Type I-like ancient solutions to the Ricci flow on non-compact 3-manifolds

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The main result of this paper is: Given any constant C, there is (ε,k,L) such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a (ε,k,L)-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the asymptotic scalar curvature ratio of a complete noncompact ancient Type I-like solution to the Ricci flow with bounded positive sectional curvature on an orientable 3-manifold, is infinity.