Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture

TL;DR

This paper corrects a mistake in the evolution equations of total curvature in Ricci flow, clarifying the dependence of bounds on initial conditions and confirming the robustness of previous extinction results.

Contribution

It provides a correction to the evolution equations in Ricci flow, specifically adjusting the dependence of bounds on initial geometric quantities.

Findings

The upper bound for total curvature evolution is exponential in time.

The multiplicative constant depends on initial total curvature and length.

The correction does not affect finite-time extinction results.

Abstract

This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-trivial.