Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

Abstract

For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.