Convergence Stability for Ricci Flow

TL;DR

This paper establishes convergence stability for Ricci flow, demonstrating that solutions starting near a stable fixed point also converge, and extends convergence results to broader classes of initial geometries without symmetry assumptions.

Contribution

It proves continuous dependence of Ricci flow on initial conditions and extends convergence results to non-symmetric initial geometries.

Findings

Proves convergence stability for Ricci flow.

Extends convergence results to open sets of initial data.

Shows stability without symmetry assumptions.

Abstract

The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state $g_0$ exists for all time and converges to a stable fixed point, then the flows of solutions that start near $g_0$ also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries.