the canonical shrinking soliton associated to a ricci flow

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1 the canonical shrinking soliton associated to a ricci flow Esther Cabezas-Rivas and Peter. Topping 15 April 2009 Abstract To every Ricci flow on a manifold over a time interval I R, we associate a shrinking Ricci soliton on the space-time I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author [17], and ccann and the second author [11]; we briefly survey the link between these subjects. 1 Introduction In 1982, Hamilton [6] introduced the study of Ricci flow, which evolves a Riemannian metric g on a manifold under the nonlinear evolution equation g t = 2 Ric(g(t)), (1.1) for t in some time interval I R. Since then, the subject has developed steadily, and has become established as an effective bridge between analysis, geometry and topology (see for example [12], [13], [14] and the overview in [16]). The initial progress relevant to the present paper was Hamilton s discovery in 1993 of the so-called Harnack quantities (see [7] for more information) and by 1995, olan Wallach [8, 14] had proposed that these quantities should arise as some sort of curvature of some higher-dimensional manifold or bundle associated to the Ricci flow. This idea was developed by Chow and Chu [1] who considered the space-time manifold I, and defined a pair ( g, ) of a metric on its cotangent bundle degenerate in the time direction and a g-compatible torsion-free connection (which is not unique owing to the degeneracy of g) so that the derivatives in the time coordinate direction of the components of ( g, ) resemble the formulae one can compute for the evolution of the components of the metric and its Levi-Civita connection under Ricci flow. (See [1] for more details.) It turns out that Hamilton s matrix Harnack quadratic is almost the Riemannian curvature of that space-time connection. An improved correspondence is established in the work of Chow and Knopf [3] by considering Ricci flow with a cosmological term. (An example of such a flow would be ḡ( t) := 1 t g(t), for t = log t, where g(t) is a Ricci flow.) In 2002, Perelman [12, 6] made a new breakthrough along these lines involving the construction of an essentially Ricci-flat manifold of dimension unbounded from above, which we now describe. The starting point is a Ricci flow which once seen with respect 1

2 to a reverse time parameter τ := C t (for some C R) is defined for τ lying in some interval I R +. Let be a large natural number, and consider the manifold := I S equipped with the metric g defined by g ij = g ij ; g 00 = 2τ + R; g αβ = τg αβ, with all remaining metric coefficients g 0i, g 0α and g iα equal to zero, where i, j are coordinate indices on the factor, α, β are those on the S factor, 0 represents the index of the time coordinate τ I, the scalar curvature is written R, and g αβ is the metric on the round S of sectional curvature 1 2. The significance of the manifold (, g) is that it is Ricci-flat up to errors of order 1. This allowed Perelman to formally apply the Bishop-Gromov comparison theorem in order to discover his reduced volume [12]. Hamilton s Harnack quantities can be recovered from the full curvature tensor, up to errors of order 1. ore recently, the theory of optimal transportation has been introduced into the study of Ricci flow, with papers by ccann and the second author [11], the second author [17] and then Lott [10]. We give more details in Section 3, but for now we mention that a notion of L-optimal transportation was introduced in [17] which can be used to recover all the important monotonic quantities for Ricci flow that were discovered by Perelman [12] in his analysis of finite-time singularities for Ricci flow (see [17] and [10]). The starting point for this paper is the proposal of John Lott that one might be able to make a formal justification of the results in [17] by applying optimal transportation theory developed for manifolds of positive Ricci curvature, directly to Perelman s construction (, g) (see also [10]). This seems to be problematic using existing optimal transportation theory. However, consideration of what alternative construction analogous to Perelman s (, g) could lie behind the results of [17] turns out to be fruitful; in this paper we are thus led to the following theorem in which we construct the canonical shrinking soliton associated to a Ricci flow on. Theorem 1.1. Suppose g(τ) is a (reverse) Ricci flow i.e. a solution of g = 2 Ric(g(τ)) defined for τ within a time interval (a, b) (0, ), on a manifold of dimension n. Suppose Ω, I (a, b), and is sufficiently large to give a positive definite metric ĝ on Ω I ˆ := (a, b) defined by ĝ ij = g ij τ ; ĝ 00 = 2τ 3 + R τ n 2τ 2 ; ĝ 0i = 0, where i, j are coordinate indices on the factor, 0 represents the index of the time coordinate τ (a, b), and the scalar curvature of g is written as R. Then up to errors of order 1, the metric ĝ is a gradient shrinking Ricci soliton on the higher dimensional space ˆ: ( ) 1 Ric(ĝ) + Hessĝ (1.2) 2τ 2ĝ, by which we mean that on Ω I ˆ, the quantity [ Ric(ĝ) + Hessĝ ( ) 1 ] 2τ 2ĝ is bounded independently of, with respect to any fixed metric on ˆ. 2

3 One aspect of this theorem is that given a Ricci flow over the time interval (a, b), it is then natural to introduce an additional time parameter and consider Ricci flow using (a, b) as the underlying manifold. This theorem serves as an example of an application of the theory of optimal transportation to prove a new result in a different field. The route between the theorems of optimal transportation and this construction is described in Sections 3 and 4. It will become apparent, from Section 4 and [17], that our canonical shrinking soliton encodes various monotonic quantities which underpin Perelman s work on Ricci flow [12, 13, 14], including entropies and quantities involving L-length (see Section 3). In Section 5 we describe a steady space-time soliton construction which will generate other monotonic quantities. Remark 1.2. Our construction also encodes Hamilton s Harnack quantities. ore precisely, the components of the (4, 0) curvature tensor τrm(ĝ) coincide (up to errors of order 1 ) with the components of the matrix Harnack expression [7]. Remark 1.3. Although each side of (1.2) evaluated on the pair (, ) will have magnitude of order, the theorem tells us that their difference does not even have any terms of order 1, only those of order 1. All errors disappear when the original Ricci flow g(τ) is a homothetically shrinking Einstein manifold, shrinking to nothing at τ = 0. Acknowledgements: We thank Robert ccann for discussions relating to an observation of Tom Ilmanen used in Section 3. Part of this work was carried out at the Centre de Recerca atemàtica, Barcelona, and the Institut Henri Poincaré, Paris, and we thank these institutions for their hospitality. This work was partly supported by the Leverhulme Trust. The first author was partly supported by DGI(Spain) and FEDER Project T The calculations In this section, we give exact formulae for the Christoffel symbols of ĝ from Theorem 1.1, and approximate formulae for its Ricci curvatures and for the Hessian of 2τ with respect to ĝ. Theorem 1.1 will then be seen to follow easily. Proposition 2.1. In the setting of Theorem 1.1, if Γ i jk are the Christoffel symbols of g(τ) at some point x, then the Christoffel symbols of ĝ at (x, τ) are given by ˆΓ i jk = Γ i jk; ˆΓi j0 = R i j δi j 2τ ; ( ˆΓ 0 jk = ĝ00 1 gjk 2τ 2 R ) jk ; ˆΓ0 τ i0 = 1 R 2τ ĝ 1 00 x i ; ˆΓi 00 = 1 R gij 2 x j ; ˆΓ0 00 = 1 2ĝ 1 00 [ 3 2τ 4 R τ 2 + R τ τ + n τ 3 ] This is a straightforward computation from the definition of the Christoffel symbols ), ˆΓ a bc := 1 2ĝad ( ĝcd x b + ĝ bd x c ĝ bc x d where a, b, c, d are arbitrary indices, and the equation of Ricci flow. Using the standard formula for the coefficients of the Ricci curvature ˆR ab = ˆΓ c ab x c ˆΓ c ac x b 3 + ˆΓ c abˆγ d cd ˆΓ d acˆγ c bd,

4 the formula for the coefficients of Hessĝ(f) ˆ 2 ab(f) = the equation for the evolution of R 2 f x a x b f x ˆΓ c c ab, R τ + R + 2 Ric 2 = 0, (see for example [16, Proposition 2.5.4]) and the contracted Bianchi identity one readily verifies the following: i R i j = 1 2 jr, Proposition 2.2. Fixing τ > 0, a time at which the Ricci flow exists, and fixing local coordinates {x i } in a neighbourhood U of some p, then in any neighbourhood V U (a, b) of (p, τ), we have ˆR ij R ij ; ˆRi0 1 2 ir; ˆR00 R τ 2 R 2τ, where denotes equality of the coefficients up to an error bounded in magnitude by C, with C > 0 a constant independent of (but depending on V and the choice of coordinates). oreover, we have ˆ 2 ij( 2τ ) g ij 2τ R ij; ˆ 2 i0 ( 2τ ) ir 2 ; ˆ 2 00 ( 2τ ) 4τ 3 + R τ + n 4τ 2 + R τ 2. By combining the formulae of Proposition 2.2 and the definition of ĝ, we deduce Theorem Optimal Transportation on Ricci flows Suppose that (, g) is a closed (compact, no boundary) Riemannian manifold, and ν 1 and ν 2 are two Borel probability measures on. For p [1, ), we define the p-wasserstein distance W p between ν 1 and ν 2 to be [ Wp g (ν 1, ν 2 ) := inf π Γ(ν 1,ν 2) ] 1 d p p (x, y)dπ(x, y), (3.1) where d(, ) is the Riemannian distance function induced by g and Γ(ν 1, ν 2 ) is the space of Borel probability measures on with marginals ν 1 and ν 2. A basic principle in the subject see Sturm and von Renesse [15] and the references therein is that two probability measures evolving under an appropriate diffusion equation should get closer in the Wasserstein sense provided the manifold satisfies some curvature condition, most famously positive Ricci curvature. In [11], ccann and the second author showed that this type of contractivity on an evolving manifold (, g(τ)) characterises super-solutions of the Ricci flow (parametrised backwards in time) by which we mean solutions to g 2 Ric(g(τ)). (3.2) 4

5 Here the relevant notion of diffusion on an evolving manifold (, g(τ)) moves the probability density u (with respect to the evolving Riemannian measure µ g(τ) ) by the parabolic equation u = g(τ)u 1 2 tr ( ) g u, and we refer to the resulting one-parameter families of measures simply as diffusions. This way, if we define a local top-dimensional form ω(τ) := u dv g(τ), then ω = g(τ)ω, (3.3) where is here the connection Laplacian, and thus the evolution of the measures corresponds to Brownian motion. The following characterisation was proved for the W 2 distance in [11]. Tom Ilmanen has pointed out to us (via Robert ccann) that this extends to the case of W 1 distance, giving: Theorem 3.1. (cf. [11, Theorem 2]) Suppose that is a closed manifold equipped with a smooth family of metrics g(τ) for τ [τ 1, τ 2 ] R. Then the following are equivalent: (A) g(τ) is a super Ricci flow (i.e. satisfies (3.2)); (B) whenever τ 1 < a < b < τ 2 and ν 1 (τ), ν 2 (τ) are diffusions (as defined above) for τ (a, b), the function τ W g(τ) 1 (ν 1 (τ), ν 2 (τ)) is weakly decreasing in τ (a, b); (C) whenever τ 1 < a < b < τ 2 and f : (a, b) R is a solution to f = g(τ)f, the Lipschitz constant of f(, τ) with respect to g(τ) is weakly increasing in τ. Proof. All implications in this theorem are proved exactly as in [11] except for (A) = (B) whose proof, pointed out by Ilmanen, we now describe. Suppose that g(τ) is a super Ricci flow, and that ν 1 (τ) and ν 2 (τ) are two diffusions, all defined for τ in a neighbourhood of τ 0 (a, b). By Kantorovich-Rubinstein duality (see e.g. [18, 1.2.1]) we have { W g(τ) 1 (ν 1 (τ), ν 2 (τ)) := max ϕdν 1 (τ) ϕdν 2 (τ) ϕ : R is Lipschitz } and ϕ Lip 1 with respect to g(τ). (3.4) Let ϕ 0 : R be a function which achieves the maximum in this variational problem at time τ 0, and extend ϕ 0 to a function ϕ : [τ 0 ε, τ 0 ] R for some ε > 0 by solving the equation ϕ = g(τ)ϕ. By the implication (A) = (C) of the theorem (proved in [11]) for all τ [τ 0 ε, τ 0 ] we have ϕ(, τ) Lip 1 and therefore ϕ(, τ) can be used as a competitor in the variational problem (3.4) to see that W g(τ) 1 (ν 1 (τ), ν 2 (τ)) ϕ(, τ)dν 1 (τ) ϕ(, τ)dν 2 (τ). But by an integration by parts formula [16, 6.3] we know that the functions τ ϕ(, τ)dν 1 (τ) and τ ϕ(, τ)dν 2 (τ) 5

6 are each independent of τ, so we deduce that W g(τ) 1 (ν 1 (τ), ν 2 (τ)) W g(τ0) 1 (ν 1 (τ 0 ), ν 2 (τ 0 )). In the next section, we will demonstrate how the manifold ( ˆ, ĝ) of Theorem 1.1 arises naturally by trying to reconcile Theorem 3.1 with the main result proved by the second author in [17], which we now rephrase into the most suggestive form for our present purposes. The idea of L-optimal transportation [17] is to transport a probability measure from one time slice of a Ricci flow to another, using a cost function derived from Perelman s L-length. ore precisely, given a time interval [τ 1, τ 2 ] (0, ) in the domain of definition of the Ricci flow, we consider the cost function c : R induced by the Lagrangian L(x, v, τ) := τ(r(x, τ) + v 2 n 2τ ) which gives τ2 ( c(x, y) = inf τ R(γ(τ), τ) + γ (τ) 2 n ) dτ, γ 2τ τ 1 where the infimum is taken over all C 1 curves γ : [τ 1, τ 2 ] for which γ(τ 1 ) = x and γ(τ 2 ) = y. Using the definitions from [12, 7] and [17] L(γ) = τ2 τ(r(γ(τ), τ) + γ (τ) 2 )dτ; Q(x, τ 1 ; y, τ 2 ) = inf τ γ 1 where the infimum is again over curves γ as above, we can write c(x, y) = Q(x, τ 1 ; y, τ 2 ) n( τ 2 τ 1 ). L(γ), This cost function then induces a distance from one Borel probability measure ν 1 (viewed as existing at time τ 1 ) to another ν 2 (viewed at time τ 2 ) via the formula D(ν 1, τ 1 ; ν 2, τ 2 ) = inf Q(x, τ 1 ; y, τ 2 )dπ(x, y) n( τ 2 τ 1 ) π Γ(ν 1,ν 2) =: V (ν 1, τ 1 ; ν 2, τ 2 ) n( τ 2 (3.5) τ 1 ), using V as defined in [17]. From the following theorem, one can recover [17] most of Perelman s monotonic quantities (both involving entropies and L-length) which are central in his work on Ricci flow [12, 13, 14]. Theorem 3.2. (Equivalent to [17, Theorem 1.1].) Suppose that g(τ) is a Ricci flow on a closed manifold over an open time interval containing [ τ 1, τ 2 ], and suppose that ν 1 (τ) and ν 2 (τ) are two diffusions (in the same sense as in Theorem 3.1) defined for τ in neighbourhoods of τ 1 and τ 2 respectively. Then the distance between the diffusions decays in the sense that for s 1 sufficiently close to 1, D(ν 1 (s τ 1 ), s τ 1 ; ν 2 (s τ 2 ), s τ 2 ) s 1 2 D(ν1 ( τ 1 ), τ 1 ; ν 2 ( τ 2 ), τ 2 ). This formulation of the theorem indicated to us that we should look for a context in which the result arises as an application of Theorem 3.1. This leads to Theorem 1.1, and we explain the connection in the next section. 4 The relationship between ( ˆ, ĝ) and L-optimal transportation In this section, as opposed to all others in this paper, we allow ourselves to make purely heuristic arguments. We present a formal argument to recover Theorem 3.2 by applying 6

7 Theorem 3.1 to the specific manifold ( ˆ, ĝ) of Theorem 1.1. While this is the simplest way to explain the link between our new manifold ( ˆ, ĝ) and optimal transportation, we stress that the main interest here is the reverse flow of ideas: the Ricci soliton ( ˆ, ĝ) was discovered by trying to find a context in which Theorem 3.1 formally implied Theorem 3.2. We begin this section by arguing formally that shortest paths in ( ˆ, ĝ) correspond to L-geodesics for the original Ricci flow. Suppose x, y and [τ 1, τ 2 ] (0, ) lies within the time domain on which the Ricci flow is defined. Consider paths Γ : [τ 1, τ 2 ] ˆ connecting (x, τ 1 ) and (y, τ 2 ) in ˆ of the form Γ(τ) = (γ(τ), τ), where γ : [τ 1, τ 2 ] satisfies γ(τ 1 ) = x and γ(τ 2 ) = y. Then τ2 Length(Γ) = γ (τ) + dτ τ 1 ĝ [ ] τ2 γ 2 1 g(τ) = + τ 1 τ 2τ 3 + R τ n 2 2τ 2 dτ τ2 [ ] 1 ( (4.1) 2 = τ 1 2τ τ 2 γ 2 + τ 2 R τn ) 2 + O( 1 ) dτ 2 = 2(τ τ ) + 1 [L(γ) n( τ 2 1 τ 1 )] + O( ). 2 3/2 Just as in [12, 6], this indicates that minimising paths Γ should essentially arise from L-geodesics γ for large, and suggests the formula dĝ((x, τ 1 ), (y, τ 2 )) = 2(τ τ )+ 1 2 [Q(x, τ 1 ; y, τ 2 ) n( τ 2 τ 1 )]+O( 1 3/2 ). (4.2) We now wish to apply Theorem 3.1 to certain diffusions on the reverse Ricci flow G(s) on ˆ := (a, b) starting at time s = 1 with the metric G(1) = ĝ. Since ĝ satisfies the (approximate) Ricci soliton equation (1.2), the theory of Ricci solitons (see [16, 1.2.2]) tells us that G(s) is (essentially) given by G(s) := s ψ s(ĝ) (4.3) where ψ s : ˆ ˆ is the family of diffeomorphisms with ψ1 the identity, obtained by integrating the vector field X := 1 s ˆ ( 2τ ) with ˆ representing the gradient with respect to ĝ. We may compute X = 2sτ 2 ĝ 1 00 and by neglecting the error of order 1 = τ s + O( 1 ), (4.4) (but see also Section 6) this integrates to ψ s (x, τ) (x, sτ). In particular, the (approximate, reverse) Ricci flow G(s) operates by pulling back ĝ in time, and scaling appropriately. Applying Theorem 3.1, we find that two diffusions on the evolving manifold ( ˆ, G(s)) should get closer in the W 1 sense as s increases. The Ricci flow g(τ) we use to generate ( ˆ, ĝ) will be that in the hypotheses of Theorem 3.2. The measures we wish to put into Theorem 3.1 will be derived from the measures ν 1 (τ) and ν 2 (τ) appearing in the hypotheses of Theorem 3.2, together with the corresponding τ 1 and τ 2. Let F τ : ˆ be defined to be the embedding F τ (x) = (x, τ); then the initial measures we wish to put 7

8 into Theorem 3.1 are (F τk ) # ν k ( τ k ), for k = 1, 2, which are each supported on a time-slice in ˆ. Because of the extreme stretching of the τ direction in the metric ĝ (recall that ĝ 00 is of order ) there is essentially no diffusion of the measures in the τ direction, and we view them as remaining supported in the time slices { τ k }. They then evolve mainly under diffusion in the factor, under the Laplacian induced by G ij (s). By (4.3) and the definition of ĝ from Theorem 1.1, on the time-slice { τ k } we have G ij (s) = s[ψ s(ĝ)] ij = s[ĝ {s τk }] ij = g ij(s τ k ) τ k. Therefore, the measures evolve by diffusion in the factor under g(s τ k ) τ k = τ k g(s τk ), which is also the evolution of s ν k (s τ k ). In other words, we have (formally) deduced that s W G(s) 1 ((F τ1 ) # ν 1 (s τ 1 ), (F τ2 ) # ν 2 (s τ 2 )) is weakly decreasing. If we now push forward this whole construction under the diffeomorphisms ψ s, and adopt the abbreviation ˆν k (τ) := (F τ ) # ν k (τ), then we find that is weakly decreasing. s W sĝ 1 (ˆν 1(s τ 1 ), ˆν 2 (s τ 2 )) = s 1 2 W ĝ 1 (ˆν 1(s τ 1 ), ˆν 2 (s τ 2 )) (4.5) ow the expansion (4.2) of the Riemannian distance on ( ˆ, ĝ) suggests that W ĝ 1 (ˆν 1(τ 1 ), ˆν 2 (τ 2 )) 2(τ τ ) D(ν 1 (τ 1 ), τ 1 ; ν 2 (τ 2 ), τ 2 ). Therefore, the monotonicity in (4.5) implies that s 2( τ τ ) + s 1 2 D(ν 1 (s τ 1 ), s τ 1 ; ν 2 (s τ 2 ), s τ 2 ) 2 is monotonically decreasing, and hence so is s s 1 2 D(ν1 (s τ 1 ), s τ 1 ; ν 2 (s τ 2 ), s τ 2 ), which is the content of Theorem 3.2 as desired. 5 Construction of a space-time steady soliton In this section we make a steady soliton construction analogous to the shrinking soliton construction of Theorem 1.1. There is also an expanding soliton construction similar to that of Theorem 1.1. Only the shrinking case is adapted to Perelman s L-length and his monotonic quantities which are so important in the study of finite-time singularities [12, 13, 14]. However, the steady case is the simplest construction of all and has its own potential applications. 8

9 Theorem 5.1. Suppose g(τ) is a (reverse) Ricci flow i.e. a solution of g = 2 Ric(g(τ)) defined for τ within a time interval (a, b) R, on a manifold of dimension n. Suppose Ω, I (a, b), and is sufficiently large to give a positive definite metric ḡ on Ω I ˆ := (a, b) defined by ḡ ij = g ij ; ḡ 00 = + R; ḡ 0i = 0, where i, j are coordinate indices on the factor and 0 represents the index of the time coordinate τ (a, b). Then up to errors of order 1, the metric ḡ is a gradient steady Ricci soliton on ˆ: in the same sense as in Theorem 1.1. Ric(ḡ) + Hessḡ ( τ) 0, (5.1) For this metric, the Christoffel symbols can be computed to be Γ i jk = Γ i jk; Γi j0 = R i j; Γi 00 = 1 R gik 2 x k ; Γ 0 jk = 1 + R R jk; Γ0 j0 = 1 2 x j ln( + R); Γ0 00 = 1 ln( + R). 2 The Ricci coefficients are, up to errors of order 1, R ij R ij ; and the coefficients of Hessḡ( τ) are Ri0 ir 2 ; R00 R τ 2, 2 ij( τ) R ij ; which yields Theorem i0 ( τ) ir 2 ; 2 00 ( τ) R τ 2, Just as the ( ˆ, ĝ) of Theorem 1.1 encodes Perelman s L-length in its geodesic distance, in the sense of (4.1), the manifold ( ˆ, ḡ) of Theorem 5.1 encodes Li-Yau s length [9] L 0 (γ) := τ2 τ 1 (R(γ(τ), τ) + γ (τ) 2 )dτ which predates L(γ), via the formula (using the same notation as in (4.1)) Length(Γ) = (τ 2 τ 1 ) L 0(γ) + O( 1 3/2 ). If one considers optimal transportation on this alternative ( ˆ, ḡ), then one recovers the variant of L-optimal transportation using the L 0 -length of Li-Yau, as studied in [10]. 6 ean curvature of time-slices In the framework of Theorem 1.1, the underlying manifold of our original Ricci flow can be regarded as a hypersurface {τ} of the ambient ( ˆ, ĝ). Its mean curvature is H = Hĝ {τ} = 1 ĝ 2 00 (R n 2τ ); in fact, the mean curvature vector Hĝ {τ} := Hν = 9

10 Hĝ from (4.4). ore precisely, we have the exact formula gives precisely the term we neglected in the approximate computation of X ˆ ( 2τ ) = τ + Hĝ {τ}. (6.1) Therefore, with X := 1 s ˆ ( 2τ ) generating diffeomorphisms ψ s as in Section 3 (with ψ 1 the identity) and G(s) := s ψ s(ĝ) (which is approximately a reverse Ricci flow) we have the following exact statement (i.e. involving no errors at all) relating reverse mean curvature flow to certain one-parameter families of time-slices {τ}, which is in a similar spirit to [2, Lemma 3.2]. Theorem 6.1. Given a flow ( ˆ, G(s)) as above, for s [1, T ), and τ 0 (a, b) a time at which the original reverse Ricci flow is defined, if we define F : [1, T ) ˆ by F(x, s) = ψs 1 (x, sτ 0 ), then the one-parameter family of submanifolds s := F( {s}) = ψ 1 s ( {sτ 0 }) is a reverse mean curvature flow within the flow ( ˆ, G(s)) of Riemannian manifolds, in the sense that F(x, s) = H s G(s) s (F(x, s)). ote that here both the (approximate) Ricci flow G(s) and the mean curvature flow s evolve in the same direction backwards with respect to s. Proof. By differentiating the expression ψ s ψs 1 that ψs 1 (ψ s ) s (x, sτ 0) = ψ s = τ 0 s (ψ 1 s = identity with respect to s, we find (x, sτ 0 )) = X(x, sτ 0, s) = 1 ( s ˆ 2τ 1 Hĝ s {sτ (x, sτ 0} 0), where we have used (6.1). Therefore, we may compute ( ψ 1 s (x, sτ 0 ) ) ( ) = ψ 1 s s s (x, sτ 0) + (ψs 1 ) τ 0 = 1 s H ψ s (ĝ) s (ψ 1 s (x, sτ 0 )) = H G(s) (F(x, s)). s ) (x, sτ 0 ) (6.2) [ 1 = (ψs 1 ) Hĝ s {sτ (x, sτ 0} 0) ] We conclude with an observation about mean curvature flow of one dimension less. Suppose that (, g(τ)) is a reverse Ricci flow for τ (a, b) and that for some (n 1)- dimensional manifold P, the map σ : P (a, b) is a reverse mean curvature flow in the sense that P τ := σ(p {τ}) is a family of smooth hypersurfaces of and σ (x, τ) = H g(τ) P τ (σ(x, τ)). Then the space-time track, which is the image of Σ : P (a, b) Σ(x, τ) := (σ(x, τ), τ), ˆ defined by is φ-minimal in ( ˆ, ĝ) for φ = 2τ, up to errors of order 1, in the sense that H φ := Hĝ Σ(P (a,b)) ˆ φ, ν Σ = O( 1 ), where ν Σ is the unit outward normal to Σ(P (a, b)) and we have used the natural generalization of mean curvature introduced in [5]. (See also Chapter of [4] for the corresponding property of Perelman s metric g described in Section 1.) 10

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