Canonical surgeries in rotationally invariant Ricci flow

TL;DR

This paper constructs and analyzes rotationally invariant Ricci flows with surgery, proving convergence to unique spacetimes and establishing curvature bounds near singularities, thus advancing understanding of symmetric Ricci flow evolution.

Contribution

It introduces a simplified construction of rotationally invariant Ricci flows with surgery and proves their convergence and stability, extending prior results to symmetric settings.

Findings

Convergence of Ricci flows with surgery to unique spacetimes.

Closeness of spacetimes measured by equivariant comparison maps.

Curvature blowup rate bounded by inverse square of remaining time.

Abstract

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.