aa r X i v : . [ m a t h . DG ] D ec CORRECTION TO ” C IN [2] IS ZERO”
Abbas Bahri
John Morgan and G,Tian pointed out a mistake in the concluding argument for our paper entitled ” C in [2]is zero”, which was recently published in arXiv:1512.02098. We hereby acknowledge this mistake and correct thecomputation, leading to the conclusion that C is non-zero and that their reference [2] does indeed fully address andresolve the counter-example which we provided in [3] to the inequality (19.10) in their monograph [1].The mistake takes place when computing ( ˆ ∇ ∂∂t ˆ ∇ S S, H ). The metric is variable here and the derivatives of theChristoffel symbols lead to a non-zero C . Namely:( ˆ ∇ ∂∂t ˆ ∇ S S, H ) = ( ˆ ∇ ˆ H ˆ ∇ S S, H ) − ( ˆ ∇ H ˆ ∇ S S, H )Now, ˆ H is along ( c ( x, t ) , t ). Thus, the metric is g ( t ) and ˆ ∇ S S = H + Ric ( S, S ) ∂∂t . Thus,( ˆ ∇ ˆ H ˆ ∇ S S, H ) = ( ˆ ∇ ˆ H ( H + Ric ( S, S ) ∂∂t ) , H ) = ( ˆ ∇ ∂∂t ( H + Ric ( S, S ) ∂∂t ) , H ) + ( ˆ ∇ H ( H + Ric ( S, S ) ∂∂t ) , H )Since, ˆ ∇ ∂∂t ∂∂t = 0 and ( ∂∂t , H ) = 0, since ( ˆ ∇ H ( Ric ( S, S ) ∂∂t ) , H ) = O ( k ),we find that: ( ˆ ∇ ∂∂t ˆ ∇ S S, H ) = ( ˆ ∇ ∂∂t H, H ) + ( ˆ ∇ H H, H ) − ( ˆ ∇ H ˆ ∇ S S, H ) + O ( k ) == ( ˆ ∇ H H, H ) − ( ˆ ∇ H ˆ ∇ S S, H ) + O ( k )Now, since S is horizontal, ˆ ∇ S S = ∇ S S + θ ∂∂t , θ bounded, so that( ˆ ∇ H ˆ ∇ S S, H ) = ˆ ∇ H ∇ S S, H ) + O ( k )Thus, our above expression is, up to O ( k ):( ∇ H H, H ) − ( ∇ H ∇ S S, H ) H ( c ( x, t ) , s ) is equal to ∇ g ( t ) S S , with S ( c ( x, t ) , s ) = ∂c ( x,t ) ∂x | ∂c ( x,t ) ∂x | g ( t ) . Along H , ( c ( x, t ) , s ) changes after the time τ into( c ( x, t + τ ) , s ). With s = t , the metric is g ( t ), so that, along a piece of curve tangent to H as defined here: ∇ S S ( c ( x, t + τ ) , s ) = ∇ g ( t ) S S , with S ( c ( x, t + τ ) , t ) = ∂c ( x,t + τ ) ∂x | ∂c ( x,t + τ ) ∂x | g ( t + τ ) instead of ∇ S S ( c ( x, t + τ ) , s ) = ∇ g ( t + τ ) S S ,with S ( c ( x, t + τ ) , t ) as above.This is the expression that we would find in ( ∇ H H, H ) and there is therefore a difference between H ( c ( x, t + τ ) , t )and ∇ g ( tS S , where S is taken at ( c ( x, t + τ ) , t ). The difference appears through the Christoffel symbols of the twodifferent metrics g ( t + τ ) and g ( t ). In ( ∇ H H, H ) − ( ∇ H ∇ S S, H ), this difference is differentiated along H , that isalong τ and it leaves a single factor for H , giving rise to C k , with C non-zero.The observations of John Morgan and Gang Tian, leading to the complete resolution of this matter, are gratefullyacknowledged here. 1 ABBAS BAHRI
References
1. J.Morgan and G.Tian,
Ricci Flow and the Poincare Conjecture , vol. 3, Clay Mathematics Monograph, AMSand Clay Institute, 2007.2. J.Morgan and G.Tian,
Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture , arXiv:1512.00699(2015).3. A.Bahri,