AN ENERGY APPROACH TO THE PROBLEM OF UNIQUENESS FOR THE RICCI FLOW

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1 AN ENERGY APPROACH TO THE PROBLE OF UNIQUENESS FOR THE RICCI FLOW BRETT KOTSCHWAR Abstract. We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu s theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this technique we prove a further uniqueness theorem for subsolutions to a general class of mixed differential inequalities and obtain an extension of Chen-Zhu s result to solutions and initial data of potentially unbounded curvature. Let = n be a smooth manifold and g 0 a Riemannian metric on. We are interested in the question of uniqueness of solutions to the initial value problem 1 gt = Rcgt, g0 = g 0, associated to the Ricci flow on. The broadest category in which uniqueness is currently known to hold in every dimension is that of complete solutions of uniformly bounded curvature. Theorem 1 Hamilton [H1]; Chen-Zhu [CZ]. Suppose g 0 is a complete metric and gt and gt are solutions to the initial value problem 1 satisfying Then gt = gt for all t [0, T ]. sup Rm gt, sup Rm gt K 0. [0,T ] [0,T ] The uniqueness of solutions to the Ricci flow is not an automatic consequence of the theory of parabolic equations, as the system 1 is only weakly-parabolic. For compact, there are two basic arguments, both due to Hamilton. The first is a byproduct of the proof of the short-time existence of solutions in Hamilton s orginal paper [H1] and is based on a Nash-oser-type inverse function theorem. The second, given in [H], effectively reduces the question of uniqueness to that for the strictly parabolic Ricci-DeTurck flow. The basis of this argument is the observation that the DeTurck diffeomorphisms, which are generally obtained as solutions to a system of ODE depending on a given solution to the Ricci-DeTurck flow, can also be represented as the solutions to a certain parabolic PDE specifically, a harmonic map heat flow which depends on the associated solution to the Ricci flow. As DeTurck s method is applicable to many other geometric evolution equations with gauge-based degeneracies, this second argument of Hamilton s gives rise to an elegant and flexible general prescription in which one exchanges the problem of uniqueness for one weakly parabolic system for the separate problems of existence and uniqueness for one or more auxiliary strictly parabolic systems. Date: ay 01. The author was supported in part by NSF grant DS /DS

2 BRETT KOTSCHWAR This latter prescription of Hamilton is also the basis of Chen-Zhu s proof [CZ] in the complete noncompact case, however its components are far from straightforward to assemble in this setting. Given the lack of general theory for the harmonic map flow into arbitrary target manifolds, the authors in particular had to solve the problem of short-time existence for their specific variant of this flow and verify the crucial accompanying estimates effectively from scratch. This they accomplished with the combination of a clever conformal transformation of the initial metric and a series of intricate a priori estimates their approach producing, as independently useful byproducts, well-controlled solutions to both the harmonic map and Ricci- DeTurck flows associated to a given solution to the Ricci flow. In this paper, however, we demonstrate that, if one is interested solely in the uniqueness of solutions to the Ricci flow, it is possible to eliminate the passage through the harmonic map and Ricci-DeTurck-flows and avoid these delicate issues of existence entirely. Given two solutions gt and gt to the initial value problem 1, our strategy is to consider a quantity of the form Et = t α g g gt + t β Γ Γ gt + Rm Rm gt Φ dµ gt for a suitable choice of the constants α and β and weight function Φ = Φx, t, and to argue from the differential inequality it satisfies that it must vanish identically. This functional E is something of a compromise between the two perhaps most obvious candidates for such a quantity: the L -norms of the differences g g and Rm Rm. Although neither g g nor Rm Rm satisfy a strictly parabolic equation, they fail to do so in such a way that their evolution equations, together with that of Γ Γ, can still be organized in a closed and for our purposes virtually parabolic system of inequalities to which the energy method may be applied. When is compact, for example, it is not hard to show that with α = β = 0, and Φ 1, we have E t CE, and hence, since E0 = 0, that E 0. When is non-compact and the curvature of gt and gt is assumed to be uniformly bounded, we can take α = 1, β 0, 1 and Φ of some sufficiently rapid decay in space in order to draw the same conclusion with much the same argument. With some minor modifications and a bit more careful estimation, we further obtain the following rather inexpensive extension of Theorem 1, which says, essentially, that uniqueness holds in the class of solutions whose curvature at times t > 0 has at most quadratic growth in the initial distance. Below r 0 x dist g0 x, x 0 is the distance with respect to the metric g 0 from some fixed x 0. Theorem. Suppose, g 0 is a complete noncompact Riemannian manifold satisfying the volume growth condition vol g0 B g0 x 0, r V 0 e V0r for some constant V 0 and all r > 0. If gt and gt are smooth solutions to 1 on [0, T ] for which 3 γ 1 g 0 x gx, t, gx, t γg 0 x, and 4 Rmx, t gt + Rmx, t gt K 0 t δ r 0x + 1, on 0, T ] for some constants γ, K 0, and δ 0, 1/, then gt = gt for all t [0, T ].

3 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 3 Note that, in particular, we do not impose any condition on the curvature tensor of the initial metric, although the volume condition is implied, for example, by a bound of the form Rcg 0 Cr 0 + 1g 0. Also observe that the uniform equivalence 3 is automatic in the case that the curvature tensors of gt and gt are assumed bounded on the time-slices {t} for t > 0 but blow-up at a rate no greater than t δ as t 0. Corollary 3. If g 0 is a complete metric satisfying and gt and gt are smooth solutions to 1 satisfying Rmx, t gt + Rmx, t gt K 0 t δ, on 0, T ] for some K 0 and δ 0, 1/, then gt = gt for all t [0, T ]. With Shi s existence theorem [S], we also have have the following special case. Corollary 4. If g 0 is complete and of bounded curvature, then any solution gt to the initial value problem 1 which remains uniformly equivalent to g 0 and obeys the bound Rmx, t K 0 r 0x + 1 on 0, T ] must have bounded curvature tensor. Other extensions of Theorem 1 and related results can be found in the papers [C], [CY], [F], [GT], [Hs], [LT], [T]. For example, in [C], Chen proves that if 3, g 0 is complete and has bounded nonnegative sectional curvature, then any two complete solutions to the Ricci flow with this initial data must agree identically; for surfaces, he proves the same result for initial data with Gaussian curvature of arbitrary sign. It is an interesting question whether his result may be extended to higher dimensions and to initial metrics with some curvature growth. Since we have only used rather classical estimates in this paper and have made no fine use of the nonlinear reaction terms in the evolution of the curvature, we expect that the combination of conditions 3 and 4 in Theorem can be relaxed further. Perhaps the general technique we describe may be of use to such future extensions as an alternative strategy to bypass the gauge-degeneracy of the equation and possibly also of use for the quantitative comparison of solutions that agree in some limiting sense as t T [,. 1. Preliminaries Going forward, we assume that gt and gt are complete solutions to 1. We will select one of the metrics, g = gt, as our reference metric and use the notation V W below to represent a linear combination of contractions of the tensors V and W and g in which the coefficients and the total number of terms are bounded by some constant depending only on the dimension. In the estimates, we will use C = Cn to such denote such a constant, which may change from line to line. To further reduce clutter in our expressions we will often use g to denote the norms induced on Tl k by g distinguishing other norms by a subscript, and simply write R = Rm and R = Rm for the 3, 1 curvature tensors associated with g and g. In the few instances in which it occurs, we will use scalg for the scalar curvature. It will be convenient also to use index notation in some of our calculations, but to potential confusion, we will only raise or lower indices explicitly to identify the metric involved. We will only violate this and then only slightly in some of

4 4 BRETT KOTSCHWAR the schematic equations below, where, according to our convention above, we would simply write, e.g., g ik g jl Tij k U k as g 1 T U, and g ij g kl T j kl as T. For similar reasons, we will only use the summation convention in its most restricted sense summing over repeated upper and lower indices from 1 to n, when it describes a metric independent contraction; for example, we can write R jk = Rljk l and R jk = R ljk l without ambiguity for the coordinate representations of Rc and Rc. Finally, we introduce the notation T max{t, 1} Evolution equations and inequalities. Define h g g, A, S R R, that is, A k ij Γk ij Γ k ij, and Sl ijk Rl ijk R ijk l. We begin by organizing the evolution equations satisfied by these tensors in such a way that every term contains either a factor of some contraction of one of the group or S. We note for later that we can write 5 g ij g ij = g ik g jl h kl, and k g ij = g aj A i ka + g ia A j ka, or, according to our convention, and 6 g 1 g 1 = g 1 h, and g 1 = g 1 A. Now, on [0, T ], the tensors h and A satisfy respectively. Since h ij = R ij R ij = S l lij Ak ij = g mk i Rjm + j Rim m Rij g mk i R jm + j R im m R ij, 7 i Rjk i Rjk = A p ij R pk + A p ik R jp, and i R jk R jk = i Sljk l, we can use the first equation in 5 to put the equation A in the schematic form for 8 A = g 1 h R + A R + S, where, according to our convention, the last term denotes a constant number of terms of contractions of S in this case, of the form i Sljk l. Similarly, using Rl ijk = Rijk l + g pq RijpR r rqk l RpikR r jqr l + RpirR l jqk r 9 g pq R ip Rqjk l + R jp Riqk l + R kp Rijq l + g pl R pq R q ijk, together with the generalization W = A W of 7 to arbitrary tensors W, we can express the evolution equation of S in the schematic form 10 S = ag ab b R g ab b R + g 1 A R + g 1 h R R + S R + S R.

5 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 5 Here the shorthand a g ab b R g ab b R represents a g ab b R l ijk gab b Rl ijk, and we have obtained 10 from 5 and 9 via the following explicit representation of the gt-laplacian of R as R l ijk = g ab a b Rl ijk = a g ab b Rl ijk = a g ab b Rl ijk A a ap g pb b Rl ijk A l ap g ab b Rp ijk + A p ai gab b Rl pjk + A p aj gab b Rl ipk + A p ak gab b Rl ijp. Now using the Cauchy-Schwarz inequality together with the above representations, we obtain that 11 h C S, A g C 1 R h + R A + S, and 1 S S div U C g 1 R A + g 1 R h + R + R S where U g ab b R g ab b R is the section of T 3 given in local coordinates by. 13 U al ijk = g ab b Rl ijk g ab b Rl ijk = g ab g ab b Rl ijk + g ab b Rl ijk b Rl ijk = g ak g bl h kl b Rl ijk + g ab A l R p bp ijk Ap R l bi pjk A p R l bj ipk A p R l bk ijp, and by div U we mean the section of T 1 3 given by div U l ijk = au al ijk. Observe that U satisfies 14 U C g 1 R h + A R. We leave these evolution inequalities in the above rather raw form for the moment and return to simplify them later in forms specialized to the specific assumptions of Theorems 1 and. 1.. A function of rapid decay. In order that our energy quantity E be welldefined and that the differentiations and integrations-by-parts we wish to perform be valid, we will need to select a weight function which decays suitably rapidly. Lemma 5. Suppose ḡt is a smooth family of complete metrics on [0, T ] satisfying γ 1 ḡ gt, where ḡ = ḡ0 and x 0. Define rx = distḡ0 x, x 0. Then, for any positive constants L 1 and L, there exists a positive constant T = T n, γ, L 1, L, T, and a function η : [0, T ] R that is smooth in t, Lipschitz and smooth dµ gt a.e. on each {t}, and that simultaneously satisfies the conditions η + L 1 η ḡt 0, and e η L r e x, on [0, τ] whenever 0 < τ T.

6 6 BRETT KOTSCHWAR Proof. We follow the construction in Chapter 1 of [CRF], cf. also [KL], [LY], and define ηx, t η B,τ x, t B r x/4τ t for on [0, τ]. The function r is continuous and smooth off of the ḡ-cut locus of x 0, where it satisfies r ḡ0 = 1, and hence r gt γ. It follows that that r is Lipschitz and smooth dµ ḡt-a.e. for each t [0, τ]. For each L 1 > 0, we have η + L 1 η ḡt B1 BL r x 1γ 4τ t, and we can guarantee the first condition provided B < 1/γL 1. t [0, τ], we have B r x/8τ ηx, t B r x/4τ, Also, for any so, given L > 0, we can ensure the second condition on [0, τ] provided 0 < τ T min{b/8l 1/, T }. The case of uniformly bounded curvature Although Theorem is strictly stronger than Theorem 1, the proof can be substantially simplified in the case of bounded curvature, and thus we give a separate argument here to demonstrate the technique. We will consider only the case of non-compact, as the proof for that of compact is nearly identical, but less involved, and can be easily reconstructed from the argument below. For the remainder of this section, we will assume that g 0, gt, and gt satisfy the assumptions of Theorem 1, and that β 0, 1 is a fixed constant..1. Derivative and decay estimates. We first recall that the global estimates of Bando [B] and Shi [S] imply that there exists a constant N = Nn, K 0, T such that 15 R gt + R gt + t R gt + t R gt + t R gt + t R gt N. on [0, T ] It is a standard argument cf., e.g., Theorem 14.1 in [H1] that the uniform curvature bounds on gt and gt imply that the metrics gt, gt, and g 0 remain uniformly equivalent. Thus, the estimates above hold for some potentially larger N when the norms are replaced by any one of gt, gt, and g0. In what follows, we will use N to denote a series of constants depending only on n, K 0, and T which may vary from one inequality to the next. We begin by noting that the same argument that yields the uniform equivalence of the metrics can be used to produce simple estimates on the decay of h and A as t 0 that imply, in particular, that t 1 h and and t β A tend to zero uniformly as t 0 on and can be continuously extended to [0, T ]. Lemma 6. Under the assumptions of Theorem 1, we have hp, t N t and Ap, t N t on [0, T ] for some constant N = Nn, K 0, T. Proof. At an arbitrary p in, we have hp, t N hp, t g0 N N t 0 t 0 Sp, s g0 ds Rp, s g0 + Rp, s g0 ds Nt,

7 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 7 and similarly, using 6, for any 0 < ɛ < t, we have Ap, t Ap, ɛ N Ap, t Ap, ɛ g0 N N t Sending ɛ 0 completes the proof. ɛ t s 1/ ds N t ɛ. ɛ Rp, s g0 + Rp, s g0 ds.. Definition and differentiability of E. Next, since the uniform curvature bound on g0 implies a lower bound on Rcg0, the Bishop-Gromov volume comparison theorem implies that vol g0 B g0 x 0, r Ne Nr for some constant N and all r > 0. Since the metrics gt and g 0 are uniformly equivalent, we thus also have vol gt B g0 x 0, r Ne Nr for some N. With any choice of B > 0 and independent of T, the function η = η B,T of Lemma 5 with ḡt = gt will satisfy e η e Br 0 /8T on [0, T ], so if we choose B > 0 sufficiently small to ensure, say, that η 3 η 0, it will still follow that any continuous function of at most sub-quadratic exponential growth in r 0 x at t will be e η dµ-integrable. So η and η, being of quadratic growth in r 0 x, are e η dµ-integrable for any t [0, T ] where, here and elsewhere, we write dµ dµ gt, as are the uniformly bounded quantities t 1 h, t β A, and S. oreover, since R = R + A R, and R = R + A R, it follows from equations 8, 10, 11, and 15 that S and a g ab b R g ab b R, and hence h, A, and S, are uniformly bounded on [ɛ, T ] for any ɛ > 0, and consequently e η dµ-integrable for each t 0, T ] although they need not be bounded as t 0. Since dµ = scalgt dµ, and the scalar curvature of gt is bounded by assumption, these observations imply that, for fixed β 0, 1 and η as above, the quantity Et t 1 h + t β A + S e η dµ is differentiable on 0, T ], and with the dominated convergence theorem satisfies lim t 0 Et = Vanishing of E. We claim in fact that E vanishes identically on [0, T ]. This is a consequence of iterating the following result. Proposition 7. There exists N 0 = N 0 n, K 0, T > 0 and T 0 = T 0 n, β 0, T ] such that E t N 0 Et for all t 0, T 0 ]. Hence E 0 on 0, T 0 ].

8 8 BRETT KOTSCHWAR Proof. For t 0, T ] and α 0, 1, define Gt S e η dµ, Ht t 1 h e η dµ, It t β A e η dµ, and J t S e η dµ, so Et = Gt + Ht + It. In view of the discussion in Section., we may freely differentiate under the integral sign and integrate by parts about any 0 < t T. As before, in the estimates below, C will denote a series of constants depending only on n, and N a series of constants depending at most on n, β, K 0, and T. Taking into account the time dependency of the norms = gt and the measure dµ = dµ gt, and using 1 together with 15 we have S G NG +, S η S e η dµ NG + S + div U, S + C g 1 R A S + C g 1 R h S + R + R S η S e η dµ NG + S + div U, S + Nt 1/ A S + N h S η S e η dµ NG + th + t β 1 I + S + div U, S η S e η dµ, for t > 0, where we have estimated Nt 1/ A S t β 1 t β A + N S and N h S tt 1 h + N S, to obtain the third line. We can then integrate by parts in the last integral to obtain S + div U, S η S e η dµ J + η S S + S U + η U S η S e η dµ J + 3 η η S + 3 U e η dµ. Here we have estimated and η S S + S U S + η S + U, η U S η S + U. Since U Nt 1 h + N A by 14 and 15, and η 3 η by assumption, putting everything together, we have 16 G NG + t + NH + t β 1 + Nt β I J NG + NH + t β 1 + NI J, for any t 0, T ], using t T, and t β T β.

9 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 9 Similarly, with 11 and 15, we compute that h H N t 1 H + t 1, h η e η dµ N t 1 H + Ct 1 S h e η dµ 17 N 1/t 1 H + CG where we have used that η 0 and Ct 1 S h 1/t h + C S Finally, using 11 and 15 again, we have A I N βt 1 I + t β, A η A e η dµ N βt 1 I + t β C g 1 R h + R A + S A e η dµ 18 N βt 1 I + NH + N βt 1 + Ct β I + J where we have again used that η Nt 1/ β h A + Ct β S A e η dµ 0 and have estimated Nt 1/ β h A + Ct β S A Nt 1 h + Ct β A + S. Combining 16, 17, and 18, we then obtain that, for any t 0, T ], E t NEt 1/t 1 Ht t 1 β t β Ct 1 β It. Thus, for T 0 sufficiently small depending only on β and C = Cn, and for some N 0 = N 0 n, K 0, β, T sufficiently large, we have E t N 0 Et on 0, T 0 ]. Since lim t 0 Et = 0, it follows from Gronwall s inequality that E 0 on [0, T 0 ]. 3. The case of potentially unbounded curvature In this section, we reduce Theorem to a special case of a general result, Theorem 13, in the following section. The strategy is essentially the same as that of the bounded curvature setting, but here we will need to organize our estimates more carefully in order to squeeze the differential inequality satisfied by our energy quantity sufficiently to absorb the growth of coefficients that we were able to regard as uniformly bounded in our previous computations Derivative estimates and consequences. First, we recall the following refined form of Shi s local first derivative estimate due to Hamilton, [H] in which the dependencies of the bound on the local curvature bound, the radius of the ball, and the elapsed time are explicit. Theorem 8 Shi/Hamilton. Suppose that, g 0 is an open Riemannian manifold in which, for a given x 0 and r > 0, the closure of B g0 x 0, r is compactly contained. Then, if gt is a solution to 1 on [0, T ] with sup R gt K 0, B g0 x 0,r [0,T ] there exists a constant C = Cn such that R gt CK 0 1 r + 1 t + K 0 1/,

10 10 BRETT KOTSCHWAR on B g0 x 0, r/ 0, T ]. We can use this result on a series of domains to obtain a derivative bound for solutions satisfying assumptions of the general form of those in Theorem. Corollary 9. Suppose, g 0 is a complete noncompact Riemannian manifold. If gt is a smooth family of complete metrics solving 1 and satisfying t δ R gt x, t K 0 r 0x + 1 on [0, T ] for some constant K and δ [0, 1], where r 0 x = dist g0 x, x 0 for some x 0, then there exists a constant K = Kn, δ, K 0, T such that 19 t δ+1/ R gt x, t Kr 0x + 1 3/. Proof. Let t 0 0, T ] and ɛ t 0 /. For any r > 0, we have Rx, t ɛ δ K 0 r + 1 on B g0 x 0, r [ɛ, T ], and so by Theorem 8, there is a C = Cn such that δ t t t δ+1/ R x, t C K 0 r + 1 ɛ r + t t ɛ + t 1/ ɛ δ K 0r + 1, on B g0 x 0, r/ ɛ, T ]. Thus, for any r 1, we have t δ+1/ 0 R x, t 0 C δ K 0 r + 1 t t 1 δ 0 δ K 0 r + 1 1/ K r + 1 3/, for some K = K n, δ, K 0, T on B g0 x 0, r/. Since t 0 0, T ] and r 1 were arbitrary, this implies that sup t δ+1/ R x, t 8K r + 1 3/ B g0 x 0,r [0,T ] for any r 1. For the pointwise estimate, we note that if x B g0 x 0, 1, we have t δ+1/ r 0 x + 13/ R x, t tδ+1/ R x, t 16K, and if x \ B g0 x 0, 1, we have t δ+1/ r0 x + R x, t t δ+1/ 13/ r0 x + 13/ This verifies 19 with K = 64K. { 8K 4r 0 x + 1 r 0 x + 1 sup R y, t y B g0 x 0,r 0x 3/ 64K. In Theorem, we assume that gt and gt are uniformly equivalent to g 0, and so, effectively, that both Rx, t g0 and Rx, t g0 have at most quadratic growth in r 0 x. With an argument exactly analogous to Lemma 6, and using 19, we could then obtain global bounds of the form hx, t Nt 1 δ r 0x + 1 and Ax, t Nt 1/ δ r 0x + 1 3/ on the decay of h and A as t 0. In fact, since the proof of Theorem 13 below is based on localized energy quantities, we will not need global estimates on either h or A, and instead we just note that since gt and gt are assumed to be smooth solutions which agree at t = 0 we have }

11 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 11 naive estimates of the form hx, t P t and Ax, t P t on Ω [0, T ] for some P = P Ω, g, g and any compact Ω. Lemma 10. For any r > 0, there exists a constant P depending on γ and the maximum values of R g0, R g0, R g0 and R g0 on B g0 x 0, r [0, T ] such that 0 sup hx, t + Ax, t P t. x B g0 x 0,r Proof. Just define P r sup R g0 + R g0 + g 1 g0 Rc g0 + g 1 g0 Rc g0. B g0 x 0,r [0,T ] Then, by 6, for any r > 0 and x B g0 x 0, r, we have h g 0 n P r and A g 0 3 P r and Ax, 0 = 0, so hx, t γ hx, t g0 nγ P t, and Ax, t γ 3/ Ax, t g0 3γ 3/ P t on Bg0 x 0, r. 3.. Evolution inequalities for h, A, and S revisited. We next organize the inequalities satisfied by the time-derivatives of t α h, t β A, and S so that the coefficients R, R, R and R are distributed across the totality of terms in a way that their growth can be adequately absorbed. Going forward, we will write ρx r 0x + 1. Lemma 11. For any solutions gt, gt to the Ricci flow on [0, T ] and any α, β, δ, σ R, there exists a constant C = Cn such that, on 0, T ], ρ t α h Cρ t α R + ρ t σ α 1 h + Cρ t t α σ S, ρ t β A Cρ t β g 1 R + R + ρ t δ g 1 + ρ t β β A t 3 and S + C t β δ R h S, S div U, S C g 1 R 4/3 A + C g 1 R 3 h 4 U C where U is as in C R /3 + R + R S, g 1 R h + R A, Proof. Since h C R h + h, h, we find that ρ t α h R Cρ t α α h + ρ h t t α, h R Cρ t α α h + Cρ t t α S h Cρ t α R + ρ t σ α h + Cρ S t tα σ

12 1 BRETT KOTSCHWAR where we have estimated ρ Cρ ρ S h tα t α t σ h + t σ S. Likewise, from 11, we have ρ t β A Cρ t β R β t Cρ t β A + A Cρ t β R R + + ρ t δ g 1 + ρ t β β t g 1 R h + R A + S, A + C t β δ R h S, where we have used ρ t β g 1 R h A C ρ t β t δ g 1 A + t δ R h, and Cρ Cρ A S tβ t β A S. Next, from 1, we have S S div U, S since C C S g 1 R A + g 1 R h + R + R S R + R + R /3 S + C g 1 R 4/3 A + C R 3 h, g 1 R A S C g 1 R 4/3 A + R /3 S, and g 1 R h S C g 1 R 3 h + R S. Finally, from 14, we have U C g 1 R h + R A. We now specialize to the setting of Theorem except that we continue to permit δ [0, 1] and define 5 α 3 + δ/, β α 1 = 1 + δ/, σ α/ = 3 + δ/4, so that, in particular, α and β, σ 1 if δ [0, 1]. Together with the derivative estimate 19, we can put the above inequalities into the following simplified form. Proposition 1. Suppose g 0 is a complete metric satisfying the volume growth condition, gt and gt are solutions to 1 satisfying 3 and the curvature bound 4 for some δ [0, 1], and α, β, σ are as in 5. Then there exists a constant N = Nn, γ, δ, K, T such that, on 0, T ], S h Nρ h t σ + S, Ā Nρ h t σ + Ā S, S div U, S Nρ h t σ + Ā + S,

13 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 13 and 9 U Nρ t σ h + Ā, where h ρt α/ h, Ā ρ 1/ t β/ A, and ρx = r 0x + 1 are as before. Proof. By assumption, we have bounds on R and R g, and from 19, estimates on R and R g. Since gt and gt are uniformly equivalent, we also have estimates on R and R, and we collect all of these estimates here with a common constant K 1 = K 1 n, δ, γ, K 0, T : sup t δ R + R K 1 ρ, sup t δ+1/ R + R K 1 ρ 3/. [0,T ] [0,T ] The assumption of uniform equivalence also implies g 1 γ g 1 g0 γ g 1 g = γ n. So now it is just a matter of substituting these bounds into 1,, 3, and 4. In what follows we will use N to denote any constant that depends only on n, δ, γ, K 1, and T. First, from 1, using σ = 3 + δ/4, we have 30 h ρ R + t α h ρ + 3+δ/4 t t ρ K t 3+δ/4 1 t 31 δ/4 + 1 h + Cρ t Nρ t h + S 3+δ/4 α 3+δ/4 S 3+δ/4 S for any t > 0, since α = 3 + δ/ and t 31 δ/4 T. Next, from, we have Ā C g 1 R + R + ρ t δ g 1 + ρ t β β t Ā + Ctα β+δ ρ R h S Nρ 1 + t 1 δ/ t 1+δ/ Ā + Nρ t δ h S Nρ 31 h + t Ā δ/ 4 S, since β δ = 1 δ/ 0, and α β + δ 1 + δ = δ. Similarly, S S div U, S { C g 1 R 4/3 A + g 1 R 3 h + R /3 + Nρ t +4δ/3 β Ā + Nρ t 3δ α h 1 + Nρ t δ/3 t δ S Nρ t 1+5δ/6 Ā + Nρ t h + Nρ 1 + t 1 δ/3 S 5δ 3/ t 1+δ/3 Nρ 3 h + t 1+δ/3 Ā + S R + R } S

14 14 BRETT KOTSCHWAR since + 4δ/3 β = 1 + 5δ/6, 3δ α = 5δ 3/, and 5δ 3 when δ [0, 1]. Finally, from 4, we have U C 1 + 5δ 6 g 1 R h + R A Nρ 1 + δ, 3 1 t 1+δ α h + 1 t δ β Ā Nρ 33 h + t Ā, 3δ 1/ since 1 + δ α = δ β = 3δ 1/. Since σ = 3 + δ/4 is the largest exponent of 1/t to appear in the coefficients of 30, 31, 3, and 33, we obtain 6, 7, 8, and A remark on the general strategy. At this point, we could multiply h, Ā, and S by suitable cutoff and decay functions and introduce localized versions of the integral quantities G, H, I, and J from the last section in order to argue as before. However, the argument we will use is only dependent on the structure of the system of inequalities in Proposition 1 and this structure is not really specific to the Ricci flow. The guiding principle behind our efforts thus far has been that, while g and g themselves do not satisfy strictly parabolic equations, the individual curvature tensors R and R do, and do so with respect to elliptic operators and whose coefficients, respectively, depend on g and g and their derivatives up to secondorder. Since the difference of g and g and those of their derivatives can be controlled in an essentially ordinary-differential way by the difference of R and R and those of their derivatives, by prolonging the system for S = R R to include just as many of the differences of g and g and their derivatives as are needed to control, we can hope to obtain a closed and nearly parabolic system of inequalities. As we have seen thanks to an integration by parts we need only to add h and A to obtain a system for which uniqueness can be established much as for strictly parabolic systems. Thus the strategy is, 1 to begin with the system of equations satisfied by the difference Rm Rm of curvature tensors, to prolong the system by lower order quantities whose vanishing is logically redundant for the purposes of establishing uniqueness, but whose inclusion allows us to account for the lack of a common elliptic operator in the separate parabolic equations satisfied by Rm and Rm, and, 3 to apply the energy method to the mixed but, in practice, nearly parabolic system of inequalities satisfied by the aggregate of the curvature and lower-order quantities. This strategy can be used to encode uniqueness problems for other geometric evolution equations, such as the mean curvature flow, into those for similar systems of inequalities, and thus we formulate a somewhat more general uniqueness result than is necessary to prove Theorem in the chance that it might be of some independent interest. This approach to handling gauge-degeneracies in evolution equations involving curvature, is similar to that employed in [K1], [K], and has its origins in the work of Alexakis [A] on unique-continuation for the vacuum Einstein equations. See also [WY].

15 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW Reduction to system of mixed differential inequalities. For the application of the result of the next section to the situation of Theorem, we ll take in the notation of that section X = T3 1, Y = T 0 T 1 and Xt = St X, Y t = ht Āt Y, and take one of the metrics, gt, as our family of reference metrics. Note that, by assumption, X is smooth on [0, T ] and satisfies Xx, 0 0 and Xx, t K 1 t δ r 0x + 1 Nt σ e Nr 0 x on 0, T ] for an appropriate constant N where σ max{3 + δ/4, δ} < 1. It is here that we need δ < 1/, as opposed to δ < 1. The family of sections Y t is smooth in t and Lipschitz over smooth but for the factors of ρ for t > 0, and, as noted in Lemma 10, satisfies Y x, t = t α ρ x hx, t + t β ρ Ax, t t α r + 1 P r on B g0 x 0, r [0, T ] for any r > 0. Thus Y x, t tends to zero uniformly as t 0 on any compact set. We will not need any assumptions on the behavior of Y t at spatial infinity. In terms of X and Y and the norms on X and Y induced by gt, Proposition 1 implies X X div U, X Y, Y Nρ t σ X + Y, 1 4 X + Nρ t σ X + Y, on 0, T ], with U Nρ/t σ X + Y. Note that, although equations 6 and 7 only directly imply an inequality on Y, in our derivation of these inequalities, we immediately estimated the contribution of the time-derivative of = gt from above by terms proportional to R Y CK 1 ρ/t δ Y Nρ/t σ Y, so we in fact also have the inequalities in the above weaker although somewhat more symmetric form. Theorem now follows at once from Theorem 13 below. 4. A uniqueness theorem for systems of virtually parabolic differential inequalities. Let, g 0 be a complete Riemannian manifold, satisfying 34 volb g0 x 0, r A 0 e A0r, for some x 0, constant A 0, and all r > 0. Define r 0 x dist g0 x 0, x and ρx r 0x + 1 as before. Suppose that gt is a smooth family of metrics on [0, T ] such that, writing g ij = p ij, the conditions 35 γ 1 g 0 gt γg 0, and t σ px, t N 0 ρx are satisfied for some constants γ, σ 0, 1, and N 0, where gt as before. Now let X = r i=1 T ki l i and Y = r i=1 T k i l represent tensor bundles i over equipped with the metrics and connections induced by gt and t, the Levi-Civita connection of gt.

16 16 BRETT KOTSCHWAR Theorem 13. For any choice of a, σ 0, 1, γ > 0, and nonnegative constants A 0, A 1, N 0, and N 1, there exists T 0 = T 0 n, γ, σ, a, A 0, A 1, N 0, N 1 > 0, such that, whenever gt is a smooth family of metrics on [0, T 0 ] satisfying 34 and 35, and Xt C X, Y t CY are families of sections depending smoothly on t 0, T 0 ] satisfying the initial conditions 36 lim sup t 0 x Ω Xx, t = 0, on every compact Ω, the growth bound lim sup t 0 x Ω 37 t σ Xx, t A 1 e A1r 0 x, on 0, T 0 ], and the system of inequalities X X div U, X 38 Y, Y on 0, T 0 ] for Ut C T X satisfying Y x, t = 0 a X + N 1ρ t σ X + Y, a X + N 1ρ t σ X + Y, 39 U N 1ρ t σ X + Y, then Xt 0, Y t 0 for all t 0, T 0 ]. Remark 14. Here, by div U = div gt U we mean the section of X whose value in the fiber over x, t 0, T 0 ] is n i=1 e i Ue i, for a gt-orthonormal basis {e i } n i=1 of T x. Remark 15. With a simple modification of the proof below and an appropriate reduction of the constant a in the statement one can substitute for the operator = gt in Theorem 13 any elliptic operator of the form L = Λ ij i j where Λt C T0 satisfies λ 1 g ij x, t Λ ij x, t λg ij x, t on 0, T 0 ] with for some constants λ and N. Λ Nρ/t σ Proof. Again, we ll assume that is non-compact, as the argument in the compact case is very similar and less involved. For the time being we will take 0 < T 0 T to be a small constant to be determined later. We begin by introducing a suitable cutoff function. Choose a nonincreasing ψ C R, [0, 1] satisfying { ψ 1 on, 1/] ψ 0 on [1, and ψ Cψ. The function φ r : [0, 1] defined by φ r x = ψr 0 x/r then satisfies φ r 1 on B g0 x 0, r/, φ r 0 on \ B g0 x 0, r, and is Lipschitz smooth off of the g 0 -cut locus of x 0. On account of the uniform equivalence of gt with g 0, we have φ r Cγr φ r off of a dµ g0 - hence dµ gt - set of measure zero in B g0 x 0, r [0, T 0 ].

17 AN ENERGY APPROACH TO UNIQUENESS FOR THE RICCI FLOW 17 Then, for any r > 0 and t > 0, we define G r X φ r e η dµ, H r Y φ r e η dµ, J r X φ r e η dµ, with E r G r + H r, where η = η B,T0 is as in Lemma 5 with B = Bn, a, γ > 0 taken small enough to ensure that 40 η 5 a 1 a η 0 on [0, T 0 ]. As noted in that lemma, this can be achieved independently of our choice of T 0, and is not affected by a further reduction of T 0. Below we will continue to use C = Cn to denote a universal constant and N any constant which depends at most on n, the ranks k i, l i, k i, l i from the definitions of X and Y, and the constants a, γ, σ, A 0, A 1, N 0, and N 1. For convenience, we ll write θ θx, t ρx/t σ. Now we compute the evolution equations for G r and H r. First, since dµ = g ij P ij dµ, taking into account the time-dependency of dµ and the norms, it follows from 35 and 38 that 41 X G rt Nθ X +, X η X φ r e η dµ Nr + 1 t σ G r + X + div U, X η X φ r e η dµ + N1 θ X + Y + a X φ r e η dµ aj r + Nr + 1 η t σ G r + H r X φ r e η dµ + X + div U, X φ r e η dµ on 0, T 0 ]. Integrating by parts in the last term in 41, we find that X + div U, X φ r e η dµ J r + and, where φ r > 0, we can estimate X U φ r + X η φ r + X φ r 1 a X φ r + X U φ r + X X + U X η φ r + φ r a U φ r + 1 a η φ r + φ r φ r e η dµ, X, and U X η φ r + φ r U φ r + η φ r + φ r X. φ r

18 18 BRETT KOTSCHWAR So, using 39 and 40, we have 4 X + div U, X η φ r e η dµ aj r + N U + 5 a 1 a η η + N X supp φ r e η dµ φr φ r aj r + Nr + 1 t σ G r + H r + N r supp φ r φ r e η dµ X e η dµ, and thus, combining 41 and 4, that, for any 0 < t T 0, G r aj r + Nr + 1 t σ E r + N 43 r X e η dµ, Ax 0,r,r/ where Ax 0, r, r/ B g0 x 0, r \ B g0 x 0, r/. Now we examine the last term in 43. Since the metric gt is uniformly equivalent to g 0, we have vol gt Ax 0, r, r/ vol g0 B g0 x 0, r γ n/ A 0 e A0r, by the volume growth assumption on g 0. Since we also assume that the integrand X satisfies the similar growth bound 37, by choosing T 0 = T 0n, γ, A 0, A 1, B sufficiently small, we can arrange that X e η dµ N ɛr T e 0, tσ Ax 0,r,r/ for some ɛ = ɛa 0, A 1, B > 0, provided T 0 T 0. So we have 44 G rt Nr + 1 t σ G r + H r aj r + N ɛr t σ r e T 0 for any r > 0 and t 0, T 0 ], if T 0 T 0. Similarly, by 38, we compute using here only that η 0 that Y H rt Nθ Y +, Y η Y φ r e η dµ Nr + 1 t σ H r + N1 θ X + Y + a X φ r e η dµ aj r + Nr t σ G r + H r. Combining 44 and 45, we conclude that, for all r > 0 and t 0, T 0 ], E rt Nr + 1 t σ E r t + N ɛr t σ r e T 0. provided T 0 T 0. It follows then, that for any 0 < t 0 < t T 0 T 0, we have ɛr T e Qrt1 σ E r t e Qrt1 σ e 0 0 E r t 0 e Qrt1 σ r r 0 e Qrt1 σ, + 1 where Qr Nr + 1/1 σ.

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