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

Abstract

For any manifold N p 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 × S q + 1 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 k > 1 there exists a solution with curvature blow-up rate ‖ R m ‖ ∞ ( t ) ≳ ( T − t ) − k \| Rm \|_{\infty } (t) \gtrsim (T-t)^{-k} with singularity modeled on a Ricci-flat cone at parabolic scales.