Erratum to "A Complete Proof of the Poincaré and Geometrization
Conjectures - Application of the Hamilton-Perelman Theory of the Ricci
Flow", Asian J. Math., Vol. 10, No. 2, 165-492, 2006
Huai-Dong Cao and Xi-Ping Zhu
We would like to thank Bruce Kleiner and John Lott for bringing to our
attention the fact that the argument concerning Claim 2 in the proof of
Perelman's singularity structure theorem (i.e., the Step 2 in the
proof of Theorem 7.1.1 in our paper, p.400-402) essentially appeared in
the initial version of their notes on Perelman's first paper posted on
in June, 2003. Thanks to this initial version of their notes, this
part of Perelman's argument had been widely understood by the experts
in the field since June of 2003.
Looking back, we realize that, sometime during the period between June
2003 and early 2004, we had looked at the initial version of Kleiner-Lott's
notes, and the argument for finite distance implies finite curvature
in our own notes back then was based on the argument in their initial
notes. More than a year later, we wrote up the Step 2 in the proof of
Theorem 7.1.1 in our paper according to our own notes and overlooked the
fact that the argument was in fact based on the work of Kleiner-Lott in
their initial notes. We apologize for failing to attribute this argument
to Kleiner and Lott in our paper due to our oversight.
In the introduction we wrote that "We would like to point out that our
proof of the singularity structure theorem (Theorem 7.1.1) is different
from that of Perelman in two aspects: (1) we avoid using his crucial
estimate in Claim 2 in Section 12.1 of ; (2) we give a new
approach to extend the limit backward in time to an ancient solution.
These differences are due to the difficulties in understanding
Perelman's arguments at these points." Regarding part (1), we acknowledge
here that our treatment, with some modifications, follows the June 2003
version of the notes of Kleiner and Lott.