A class of Riemannian manifolds that pinch when evolved by Ricci flow

Abstract:

The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N ⁿ is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and

where K is some compact set .