  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.
