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11) is n M (∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV n = M − (∆f − |∇f |2 )(2τ ∆f − τ |∇f |2 )(4πτ )− 2 e−f dV n M n (−∇i f )(∇i R)(4πτ )− 2 e−f dV M n 2 =τ M − |∇f |2 (4πτ )− 2 e−f dV + τ (−∇i f )(∇i (2∆f − |∇f | ))(4πτ )− 2 e−f dV n M ∆f (4πτ )− 2 e−f dV − 2τ n M ∇i f ∇j Rij (4πτ )− 2 e−f dV n = −2τ M (∇i f )(∇i ∆f − ∇f, ∇i ∇f )(4πτ )− 2 e−f dV M [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV n + 2τ + 2τ M = −2τ M − 1 n gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ [(∇i f ∇j f − ∇i ∇j f )∇i ∇j f − Rij ∇i f ∇j f n − ∇i ∇j f ∇i f ∇j f ](4πτ )− 2 e−f dV n + 2τ M [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV 1 n gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ M n 1 = 2τ (∇i ∇j f ) ∇i ∇j f + Rij − gij (4πτ )− 2 e−f dV.
Dt THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 205 On the other hand, for every t, − d 1 log V = dt V M RdV ≥ λ(gij (t)) by the definition of λ(gij (t)). It follows that on an expanding breather on [t1 , t2 ], ¯ ij (t)) = λ(gij (t))V n2 (gij (t)) < 0 λ(g for some t ∈ [t1 , t2 ]. Then by using statement (i), it implies ¯ ij (t1 )) < λ(g ¯ ij (t2 )) λ(g ¯ ij (t)) is invariant unless we are on an expanding gradient soliton. We also note that λ(g under diffeomorphism and scaling which implies ¯ ij (t1 )) = λ(g ¯ ij (t2 )).
In [60], Hamilton obtained the following Li-Yau estimate for the scalar curvature R(x, t). 2 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface M . Assume the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then the scalar curvature R(x, t) satisfies the Li-Yau estimate ∂R |∇R|2 R − + ≥ 0. 4) Proof. By the above discussion, we know R(x, t) > 0 for t > 0. 4) is equivalent to ∂L 1 1 − |∇L|2 + = △L + R + ≥ 0. 5) Q= ∂L − |∇L|2 = △L + R.