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Also note λ(gij ) is invariant under diffeomorphism. 4. (i) λ(gij (t)) is nondecreasing along the Ricci flow and the monotonicity is strict unless we are on a steady gradient soliton; (ii) A steady breather is necessarily a steady gradient soliton. To deal with the expanding case we consider a scale invariant version ¯ ij ) = λ(gij )V n2 (gij ). λ(g Here V = V ol(gij ) denotes the volume of M with respect to the metric gij . -D. -P. 5. ¯ ij ) is nondecreasing along the Ricci flow whenever it is nonpositive; more(i) λ(g over, the monotonicity is strict unless we are on a gradient expanding soliton; (ii) An expanding breather is necessarily an expanding gradient soliton.

Whenever ϕ(t) ≥ 0 on [a, b), For arbitrary ε > 0, we shall show ϕ(t) ≤ ε(t − a) on [a, b]. Clearly we may assume ϕ(a) = 0. Since lim sup h→0+ ϕ(a + h) − ϕ(a) ≤ 0, h there must be some interval a ≤ t < δ on which ϕ(t) ≤ ε(t − a). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 219 Let a ≤ t < c be the largest interval with c ≤ b such that ϕ(t) ≤ ε(t − a) on [a, c). Then by continuity ϕ(t) ≤ ε(t − a) on the closed interval [a, c]. We claim that c = b. Suppose not, then we can find δ > 0 such that ϕ(t) ≤ ε(t − a) on [a, c + δ] since lim sup h→0+ ϕ(c + h) − ϕ(c) ≤ 0.

More applications will be given in Chapter 5. Let M be a complete manifold equipped with a one-parameter family of Riemannian metrics gij (t), 0 ≤ t ≤ T , with T < +∞. Let V → M be a vector bundle with a time-independent bundle metric hab and Γ(V ) be the vector space of C ∞ sections of V . e. ∆ (∇t )i hab = (∇t ) ∂ ∂xi hab = 0, ∂ ∂ for any local coordinate { ∂x 1 , . . , ∂xn }. The Laplacian ∆t acting on a section σ ∈ Γ(V ) is defined by ∆t σ = g ij (x, t)(∇t )i (∇t )j σ. For the application to the Ricci flow, we will always assume that the metrics gij (·, t) evolve by the Ricci flow.