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Analytical Geometry by Siceloff L.P., Wentworth G., Smith D.E.

By Siceloff L.P., Wentworth G., Smith D.E.

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P. 1) which starts in Z at t0 will remain in Z for all later times; (ii) ϕ + N (ϕ, t) ∈ Tϕ Z for all ϕ ∈ ∂Z and t ∈ [0, T ). Proof. We say that a linear function l on Rn is a support function for Z at ϕ ∈ ∂Z and write l ∈ Sϕ Z if |l| = 1 and l(ϕ) ≥ l(η) for all η ∈ Z. Then ϕ+N (ϕ, t) ∈ Tϕ Z if and only if l(N (ϕ, t)) ≤ 0 for all l ∈ Sϕ Z. Suppose l(N (ϕ, t)) > 0 for some ϕ ∈ ∂Z and some l ∈ Sϕ Z. 1) cannot remain in Z. To see the converse, first note that we may assume Z is compact. This is because we can modify the vector field N (ϕ, t) by multiplying a cutoff function which is everywhere nonnegative, equals one on a large ball and equals zero on the complement of a larger ball.

Vn−l+1 } at x. Let us evolve ρ(x) by the heat equation ∂ ρ = ∆ρ ∂t with the Dirichlet condition ρ|∂Ω = 0 to get a smooth function ρ(x, t) defined on Ω×[0, T ]. By the standard strong maximum principle, we know that ρ(x, t) is positive everywhere in Ω for all t ∈ (0, T ]. For every ε > 0, we claim that at every point (x, t) ∈ Ω × [0, T ], there holds n−l+1 Mαβ (x, t)viα viβ + εet > ρ(x, t) i=1 for any (n − l + 1) orthogonal unit vectors {v1 , . . , vn−l+1 } at x. We argue by contradiction.

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 )). λ(g Therefore the breather must be an expanding gradient soliton. 5 imply that all compact steady or expanding Ricci solitons are gradient ones.

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