$C_1$ in [2] is zero

TL;DR

This paper proves that the constant $C_1$ in a key lemma of a Ricci flow correction can be zero, confirming the validity of a previous counterexample and highlighting deeper issues in the original monograph.

Contribution

It establishes that the constant $C_1$ in Morgan and Tian's correction can be zero, clarifying the validity of prior counterexamples and addressing a deeper problem in the original work.

Findings

The constant $C_1$ can be taken as zero.

The previous counterexample remains valid after correction.

The division by $k$ in the original monograph was unjustified.

Abstract

In this short Note, we establish that the constant $C_1$ in Lemma $0.4$ of the correction (Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture, arXiv/math/DG:1512.00699 (2015)) by John Morgan and Gang Tian to their Clay Institute Monograph (Ricci Flow and the Poincare Conjecture, vol. 3, Clay Mathematics Monograph, AMS and Clay Institute, (2007)) can be taken to be zero. This implies that the counter-example which we provided recently in ArXiv/math/DG:1422760 still stands after this correction. Let us observe that the unjustified division by $k$ in the original monograph-the only mistake which we could find-has been addressed in the correction, so that the problem (according to our understanding and after this Note) lies deeper. We would like to thank John Morgan, Terry Tao and Gang Tian for a thorough mathematical exchange of arguments. The various conclusions have now been made available to a wider community of colleagues. Appropriate adjustments in the understanding will hopefully follow.