A relative entropy and a unique continuation result for Ricci expanders
aa r X i v : . [ m a t h . DG ] J a n A RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCIEXPANDERS
ALIX DERUELLE AND FELIX SCHULZE
Abstract.
We prove an optimal relative integral convergence rate for two expanding gra-dient Ricci solitons coming out of the same cone. As a consequence, we obtain a uniquecontinuation result at infinity and we prove that a relative entropy for two such self-similarsolutions to the Ricci flow is well-defined.
Contents
1. Introduction 12. Setup 53. Integral decay estimates 134. Preliminary integral bounds 255. Frequency bounds 306. Frequency decay 417. Decay estimates and traces at infinity 448. A relative entropy for Ricci gradient expanders 52References 621.
Introduction
Overview. A Ricci soliton is a triple ( M n , g, X ) where ( M n , g ) is a Riemannian manifoldand a vector field X satisfying the equationRic( g ) − L X ( g ) + λ g = 0 , for some λ ∈ {− , , } . We call X the soliton vector field . A soliton is said to be steady if λ = 0, expanding if λ = 1, and shrinking if λ = −
1. Moreover, if X = ∇ g f for somereal-valued smooth function f on M called the potential function then ( M n , g, ∇ g f ) is saidto be a gradient soliton. In this paper, we focus on expanding gradient Ricci solitons whoseequation reduces to 2 Ric( g ) + g = L ∇ g f ( g ) . (1.1)Notice that equation (1.1) normalizes the metric and defines the potential function f up toan additive constant.A Ricci soliton is said to be complete if ( M n , g ) and the vector field ∇ g f are complete inthe usual sense. By [Zha09], the completeness of ( M n , g ) implies the completeness of ∇ g f . Toeach expanding gradient Ricci soliton ( M n , g, ∇ g f ), one may associate a self-similar solutionof the Ricci flow as follows: g ( t ) = tϕ ∗ t g, where ( ϕ t ) t> is the one-parameter family of diffeomorphisms generated by the vector field −∇ g f /t such that ϕ t =1 = Id M . This solution is Type III , i.e. there exists a nonnegativeconstant C such that for any t ∈ (0 , + ∞ ), t sup M | Rm( g ( t )) | ≤ C, if the curvature is bounded on the manifold M n . Therefore, it is likely that expandinggradient Ricci solitons are good candidates for singularity models for Type III solutions tothe Ricci flow. To illustrate these heuristics more accurately, let us mention that the secondauthor and Simon [SS13] have shown that expanding gradient Ricci solitons naturally ariseas a blow-down of non-compact non-collapsed Type III solutions with non-negative curvatureoperator.Given an expanding gradient Ricci soliton ( M n , g, ∇ g f ) with quadratic Ricci curvaturedecay together with covariant derivatives, one can associate a unique tangent cone ( C ( S ) , dr + r g S , o ) with a smooth Riemannian link ( S, g S ): [Sie13, CD15, Der17a]. In particular, suchan expanding gradient Ricci soliton is asymptotically conical . Moreover, the metric cone( C ( S ) , dr + r g S , o ) can be interpreted as the initial condition of the Ricci flow ( g ( t )) t> associated to the soliton in the sense that ( M n , d g ( t ) , p ) t> converges to ( C ( S ) , dr + r g S , o )in the pointed Gromov-Hausdorff sense as t →
0, if p is a critical point of the potentialfunction f .We refer to Yai’s recent work [Lai20] for an application of the existence of continuousfamilies of positively curved asymptotically conical expanding gradient Ricci solitons, basedon the work of the first author [Der16], to show the existence of certain collapsed steadygradient Ricci solitons.In this article we investigate the uniqueness question among the class of asymptoticallyconical expanding gradient Ricci solitons coming out of a given metric cone over a smoothlink.Let us mention first some basic examples of such self-similar solutions. A fundamentalexample in all dimensions is the Gaussian soliton ( R n , δ eucl , r∂ r / J ( ∇ g f ) is a Killing field, where J is the natural almost complexstructure associated to any surface. Therefore, the soliton equation reduces to an ordinarydifferential equation. It is shown in [CLN06, Chapter 4] that there is a unique one-parameterfamily g c of such expanding gradient Ricci solitons, each metric g c being K¨ahler and asymp-totic to a Euclidean cone of angle c ∈ (0 , π ). The first examples in higher dimensions aredue to Bryant: Chodosh [Cho14] proved that any expanding gradient Ricci soliton with pos-itive sectional curvature and asymptotic to ( C ( S n − ) , dr + ( cr ) g S n − , o ), c ∈ (0 , C n and Feldman-Ilmanen-Knopf [FIK03] by extending Cao’sansatz have shown the existence of K¨ahler expanding gradient Ricci solitons living on the totalspace of the tautological line bundles L − k , k > n , over CP n − . These expanding solitons are U ( n )-invariant and asymptotic to the rotationally symmetric cone (cid:0) C ( S n − / Z k ) , i∂ ¯ ∂ | z | p /p (cid:1) ,the aperture p > RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 3
Observe that the uniqueness question is of interest even in the case of an asymptotic cone( C ( S ) , dr + r g S , r∂ r /
2) which is Ricci flat and endowed with the radial vector field r∂ r / C ( S ) , dr + r g S , r∂ r /
2) then uniqueness ofthe Ricci flow fails in case metric singularities are allowed. Now, even if we restrict ourattention to complete expanding gradient Ricci solitons coming out of a given Ricci flat cone,Angenent-Knopf [AK19] have proved that uniqueness still fails for some Ricci flat cones indimension greater than 4.1.2.
Main results.
The first main result of this paper is the following unique continuationresult at infinity for two expanding gradient Ricci solitons coming out of the same cone.
Theorem 1.1 (Informal statement of the main unique continuation result) . Let ( M n , g , ∇ g f ) and ( M n , g , ∇ g f ) be two expanding gradient Ricci solitons coming out of the same cone ( C ( S ) , g C := dr + r g S , r ∂ r ) over a smooth link ( S, g S ) . Assume the soliton metrics g and g are gauged in such a way that their soliton vector fields coincide outside a compact set.Then the trace at infinity lim r → + ∞ r n e r ( g − g ) =: tr ∞ (cid:16) r n e r ( g − g ) (cid:17) exists in the L ( S ) -topology, it preserves the radial vector field ∂ r and its tangential part isdivergence free with respect to the metric on the link in the weak sense. Moreover, g and g coincide pointwise outside a compact set if and only if their associated trace at infinityvanishes, i.e. tr ∞ (cid:16) r n e r ( g − g ) (cid:17) ≡ . The main tool to show the above result is as follows: we establish the existence of a suit-able frequency function at infinity, where the method follows the work of Bernstein for meancurvature flow in codimension one [Ber17], which itself is based on the fundamental work ofGarofalo-Lin [GL87]. The main difficulty and crucial point in this approach comes from thefact that different to the case of mean curvature flow, where the graphical representation atinfinity of one expander over the other yields a natural well-controlled gauge, in the currentsetting it is necessary to establish a suitable gauge at infinity between the two expanders.To establish the needed decay estimates for the frequency function it is necessary to simul-taneously control the gauge together with the frequency function. The gauge we employ isa Bianchi gauge, motivated by the work of Kotschwar [Kot17] for the comparison of twogeneral solutions of Ricci flow. Due to self-similarity of our solutions the evolution equationfor the Bianchi gauge turns into an ODE, which then results in an ODE-PDE system for thefrequency function set-up.Kotschwar-Wang [KW15] have employed Carleman estimates to prove the uniqueness ofRicci shrinkers smoothly asymptotic to a smooth cone. We expect that similar to work ofBernstein [Ber17] for mean curvature flow, the methods in this paper can be adapted to givean alternative proof of the result of Kotschwar-Wang. But different to the case treated byBernstein, the setup for Ricci shrinkers does not directly transform in the system treatedin the current paper. A unique continuation result for expanders asymptotic to Ricci flatcones was obtained by the first author using Carleman estimates [Der17b]. The results ofBernstein for mean curvature flow have been extend to the higher codimension case by Khan[Kha20]. The unique continuation result of Bernstein [Ber17] has been employed centrally by
ALIX DERUELLE AND FELIX SCHULZE
Bernstein-Wang [BW17] in their proof that the space of expanders smoothly asymptotic tosmooth cones has the structure of a smooth Banach manifold. Frequency bounds for solutionsto a general class of drift laplacians equations have been obtained by Colding-Minicozzi in[CM18].In case the asymptotic cone is Ricci flat, the convergence rate was shown to hold pointwisein the smooth sense in [Der17b]. For an arbitrary asymptotic cone, Theorem 1.1 shows thatthe same convergence rate holds for the L norm on level sets of the distance function fromthe tip of the cone.As an application of the decay estimates achieved via the frequency function, we show theexistence of a relative entropy for two expanders asymptotic to the same cone. Theorem 1.2 (A relative entropy for two expanders coming out of the same cone) . Let ( M n , g , ∇ g f ) and ( M n , g , ∇ g f ) be two expanding gradient Ricci solitons coming out ofthe same cone ( C ( S ) , g C := dr + r g S , r ∂ r ) over a smooth link ( S, g S ) . Then the followinglimit exists for all t > and is constant in time: W rel ( g ( t ) , g ( t )) := lim R → + ∞ (cid:18) ˆ f ( t ) ≤ R e f ( t ) (4 πt ) n dµ g ( t ) − ˆ f ( t ) ≤ R e f ( t ) (4 πt ) n dµ g ( t ) (cid:19) . (1.2)Feldman-Ilmanen-Ni [FIN05] have introduced a forward reduced volume and an expandingentropy (denoted by W + ) that detect expanding gradient Ricci solitons on a closed manifold.The purpose of Theorem 1.2 is to provide a replacement of the aforementioned functionals inthe non-compact setting.In order to prove that the limit (1.2) in Theorem 1.2 is well-defined, we invoke the inte-gral convergence rate for the difference of the soliton metrics g − g obtained in Theorem1.1. Observe that comparing the solutions to their common initial cone metric only yields aquadratic decay and is therefore not sufficient to ensure the existence of the limit (1.2). Weunderline the fact that (1.2) is established by taking differences rather than by considering arenormalization.For mean curvature flow, the authors established the existence of a relative entropy forexpanders asymptotic to the same cone in [DS20]. For the harmonic map heat flow the firstauthor established a similar result in [Der19]. Unlike the proof of the aforementioned works,we underline the fact that the main difficulty in proving Theorem 1.2 lies in the absence ofpointwise estimates on the difference of the soliton metrics g − g and its derivatives, usuallyobtained via the maximum principle. This partially motivates the technical effort needed toprove Theorem 1.2 in Section 8.For an extension to general solutions coming out of a cone in the case of mean curvatureflow, see the paper by Bernstein-Wang [BW19]. In future work, we will extend the aboveresult to general Type III solutions of the Ricci flow coming out of a cone.1.3. Outline of paper.
We begin in Section 2 by recalling the basics of expanding gradientRicci solitons, the soliton identities that we use all along the paper, as well as the asymptoticgeometry of self-similar solutions of the Ricci flow coming out of a metric cone over a smoothlink. Proposition 2.6 and Corollary 2.7 reduce the proof of Theorem 1.1 to the analysis of acoupled system of ODE-PDE equations. This lets us define the boundary and flux functionsassociated to such a system: Lemmata 2.8 and 2.9 start estimating the variation of theboundary function as in [Ber17].In Section 3, we analyse the above mentioned coupled system of ODE-PDE equations atthe energy level to ensure that all relevant tensors do belong in a suitable weighted L space RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 5 as soon as two expanding soliton metrics are coming out of the same cone. This legitimatesthe definition of the rescaled energy of the difference of two soliton metrics coming out of thesame cone over ends of the cone and it validates all the integration by parts in the remainingsections. Here, we proceed as in [Ber17, Section 9] with the difference that the Bianchi one-form needs to be estimated in terms of the difference of the two soliton metrics: see Lemmata3.3 and 3.5. This cumulates in the statements of Theorem 3.6 and Corollary 3.10.Section 4 proves that the two frequency functions related to the energy and the flux of thedifference of two soliton metrics are well-defined unless the soliton metrics are equal: this is thecontent of Corollary 4.4. Moreover, Lemma 4.3 shows these two frequency functions controleach other whenever they are well-defined. Finally, Lemma 4.5 establishes a crucial qualitativePohozaev type estimate whose proof relies on a delicate integration by parts involving boththe difference of the soliton metrics and the Bianchi one-form.We start estimating the variation of the frequency functions in Section 5. Proposition5.1 takes care of the generic variation of the frequency function related to the flux througha so called Rellich-Necas identity adapted to C loc symmetric 2-tensors. Based on Lemma4.5, this frequency function is shown to satisfy a first order differential inequality: this is thecontent of Corollary 5.5. The variation of the frequency function related to the energy is moresubtle because of the presence of the L norm of the Lie derivative of the Bianchi one-formon spheres of large radii. To circumvent this issue, an integration by parts is performed onsuch spheres thanks to Lemma 5.3 and culminates in the statement of Proposition 5.4. Thisfrequency function satisfies in turn a first order differential inequality modulo the derivativeof the frequency function associated to the Bianchi one-form, the latter being taken care byProposition 5.2.In Section 6, the frequency function ˆ N associated to the energy of the difference of twosuch soliton metrics is shown to decay sub-quadratically: this is the content of Theorem 6.3.The almost optimality of this decay is due to the presence of the Bianchi one-form. As a pre-liminary step, Theorem 6.1 first asserts that ˆ N merely tends to 0 at infinity then Proposition6.2 starts discretizing the bootstrapping technique needed for the proof of Theorem 6.3.With the results of Section 6 in hand, Theorem 1.1 is proved in Section 7: see Theorem7.1 for a rigorous statement. Propositions 7.6 and 7.7 establish further properties of thetrace at infinity of the difference of two soliton metrics coming out of the same cone. Theirproofs would be almost straightforward if such trace at infinity (and its convergence to it)was smooth. These statements find their counterparts for K¨ahler expanding gradient Riccisolitons coming out of a cone in Section 7.2.In the last Section 8, Theorem 8.2 establishes the existence of a relative entropy betweentwo expanding gradient Ricci solitons coming out of the same cone. Its proof relies on theintegral convergence rate obtained in Section 7 for the decay of the difference of two suchsoliton metrics only. The unique continuation result is not needed here.1.4. Acknowledgements.
The first author was supported by grant ANR-17-CE40-0034 of theFrench National Research Agency ANR (Project CCEM). The second author was supportedby a Leverhulme Trust Research Project Grant RPG-2016-174.2.
Setup
A gauge at infinity for expanding gradient Ricci solitons.
ALIX DERUELLE AND FELIX SCHULZE
In this section, we explain the results obtained in [CDS19] to identify the soliton vectorfield associated to an asymptotically conical expanding gradient Ricci soliton to the radialvector field r ∂ r on the tangent cone at infinity.A Riemannian manifold ( M n , g ) has quadratic curvature decay with derivatives if for some(hence any) point p ∈ M and all k ∈ N , A k ( g ) := sup x ∈ M | ( ∇ g ) k Rm( g ) | g ( x ) d g ( p, x ) k < ∞ , where Rm( g ) denotes the curvature of g and d g ( p, x ) denotes the distance between p and x with respect to g .Finally, to each expanding gradient Ricci soliton ( M n , g, ∇ g f ), consider the one-parameterfamily ( ϕ t ) t> of diffeomorphisms generated by − ∇ g ft and define as in Siepmann [Sie13] thefollowing sets, for any a ≥ M a := n x ∈ M | lim inf t → + tf ( ϕ t ( x )) > a o . These sets are invariant under the diffeomorphisms ( ϕ t ) t ∈ (0 , . With these definitions in hand, we can state the main result we need in this paper:
Theorem 2.1 (Map to the tangent cone for expanding Ricci solitons) . Let ( M n , g, ∇ g f ) bea complete expanding gradient Ricci soliton with quadratic curvature decay with derivatives.Then on each end of M ,(1) the one-parameter family of functions ( tf ◦ ϕ t ) t> converges to a non-negative con-tinuous real-valued function q ( x ) := lim t → + tf ( ϕ t ( x )) pointwise on M as t → + .Moreover, the convergence is locally smooth on M .(2) The metrics ( g ( t )) t> converge to a Riemannian metric ˜ g in C ∞ loc ( M ) as t → + and ˜ g = 2 Hess ˜ g q .(3) The function q is proper and satisfies on M , |∇ ˜ g q | g = q and ∇ ˜ g q = ∇ g f . Inparticular, there exists c > such that the level sets q − { c } are compact connectedsmooth hypersurfaces for c ≥ c . (4) Define ρ := 2 √ q on M and let S := ρ − ( { c } ) for any c large enough. Then thereexists a diffeomorphism ι : ( c, ∞ ) × S → M c such that g := ι ∗ ˜ g = dr + r g S c and dι ( r∂ r ) = 2 ∇ g f , where r is the coordinate on the ( c, ∞ ) -factor and g S is therestriction of ˜ g to S . Moreover, along M c , we have that | ( ∇ g ) k ( ι ∗ g − g − g )) | g = O ( r − − k ) for all k ∈ N . The statement and the proof of Theorem 2.1 is the combination of the statements and theproofs contained in [CDS19, Section 3 . Remark 2.2.
In the sequel, we apply Theorem 2.1 to two expanding gradient Ricci solitons ( M n , g i , ∇ g i f i ) i =1 , with quadratic curvature decay with derivatives which come out of thesame cone. In particular, with the notations of Theorem 2.1, there exist two diffeomorphisms ι i , i = 1 , , such that lim t → + tι ∗ i ϕ it ∗ f i = r on the end of the common metric cone where ( ϕ it ) t> is the flow generated by −∇ g i f i /t and | ( ∇ g ) k ( ι ∗ g − ι ∗ g ) | g = O ( r − − k ) for all k ∈ N . (2.1) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 7
Soliton identities.
We consider an asymptotically conical expanding gradient Ricci soliton ( M n , g, ∇ g f ). Werecall the following fundamental identities ∇ g, f = Ric( g ) + g , (2.2)∆ g f = R g + n , (2.3) ∇ g R g +2 Ric( g )( ∇ g f ) = 0 , (2.4) |∇ g f | g + R g = f, (2.5)div g Rm( g )( Y, Z, U ) = Rm( g )( Y, Z, ∇ g f, U ) , (2.6)for any vector fields Y , Z , U whose proofs can be found in [CLN06, Chapter 4] for instance.Identity 2.5 holds up to an additive constant in general. This choice normalizes the potentialfunction f .In the sequel, an expanding gradient Ricci soliton is said to be normalized if (2.5) holds on M .Now, outside of a compact set K ⊂ M we have that f is smooth and positive and we candefine r := 2 p f . We note that (2.2) implies that for r ≥ R and R sufficiently large there exists Λ > (cid:12)(cid:12) ∇ g, r − g (cid:12)(cid:12) g ≤ Λ r ≤ / , (2.7)as well as (2.5) implies that ||∇ g r | g − | ≤ Λ r ≤ / , (2.8)where we used the quadratic decay of the curvature along the end as well as the existence ofa C > r − C ≤ f ≤ r C where r is the radial coordinate along the asymptotic cone C ( S ). Moreover, we record thefollowing estimates that follow essentially from (2.7): (cid:12)(cid:12)(cid:12)(cid:12) div g ∇ g r − n − r (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∆ g r − n − r (cid:12)(cid:12)(cid:12)(cid:12) ≤ Λ r . (2.9)Then we see that we have what is denoted in [Ber17] a ’weakly conical end’. We also definethe vector fields ∂ r = ∇ g r , N = ∇ g r |∇ g r | and X = r ∇ g r |∇ g r | . For further notation and basic estimates we refer to [Ber17, Section 2].We now define Φ m = r m e − r = 2 m f m e − f . The self-adjoint operator with respect to this weight is L m = ∆ g − r ∇ g ∇ g r + m r ∇ g ∇ g r =: ∆ − f + m r ∂ r . ALIX DERUELLE AND FELIX SCHULZE
Similarly we consider Ψ m = r m e r = 2 m f m e f . Note that the self adjoint operator with respect to this weight is given by L + m = ∆ g + r ∇ g ∇ g r + m r ∂ r =: ∆ f + m r ∂ r . Finally, we recall the link between the Lie derivative of a tensor T on M along a smoothvector field X on M and the covariant derivative of T along X : Lemma 2.3.
Let ( M n , g ) be a Riemannian manifold and let X be a smooth vector field on M .If T is a tensor of type ( p, on M , then: L X T ( Y , ..., Y p ) = ∇ gX T ( Y , ..., Y p ) + p X i =1 T ( Y , ..., Y i − , ∇ gY i X, Y i +1 , ..., Y p ) , where Y i , i = 1 , ..., p , are smooth vector fields on M . In particular, if T is a (2 , -tensor andif X = ∇ g f for some smooth function f : M → R , then, L ∇ g f T = ∇ g ∇ g f T + T ◦ ∇ g, f + ∇ g, f ◦ T. (2.10)2.3. An ODE-PDE system.
Let ( M n , g , ∇ g f ) and ( M n , g , ∇ g f ) be two expanding gradient Ricci solitons smoothlycoming out of the same metric cone ( C ( S ) , dr + r g X , r∂ r /
2) such that outside a compactset, ∇ g f = r∂ r = ∇ g f . This is legitimated by Theorem 2.1 up to a diffeomorphism atinfinity. We say that the soliton metrics g and g are gauged with respect to the solitonvector field.If h := g − g , we define a one parameter family of metrics ( g σ ) σ ∈ [1 , and a one parameterfamily of potential functions ( f σ ) σ ∈ [1 , on a fixed end of M as follows: g σ := (2 − σ ) g + ( σ − g = g + ( σ − h, f σ := (2 − σ ) f + ( σ − f . Moreover, we define the linearized Bianchi one-form associated to g and g by: B := div g h − g ( ∇ g tr g h, · ) . (2.11)Finally, define the action of the curvature tensor on symmetric 2-tensors on M as follows :(Rm( g ) ∗ h ) ij := Rm( g ) iklj h kl , for h ∈ S T ∗ M. We record the following lemma that sums up the qualitative properties of the solitonidentities from the previous section applied to the family of approximate Ricci expandingsolitons with metric g σ and potential function f σ for σ ∈ [1 ,
2] we will use in the next sections:
Lemma 2.4.
Under the above assumptions and notations, define r σ := 2 √ f σ , σ ∈ [1 , . Thenthe following estimates hold true and are uniform in σ ∈ [1 , : | f σ − f | ≤ C, (2.12) (cid:12)(cid:12)(cid:12) ∇ g σ , f σ − g σ (cid:12)(cid:12)(cid:12) g σ ≤ C r − σ , (2.13) (cid:12)(cid:12)(cid:12) ∆ g σ f σ − n (cid:12)(cid:12)(cid:12) ≤ C r − σ , (2.14) | Ric( g σ )( ∇ g σ f σ ) | g σ ≤ C r − σ , (2.15) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 9 |∇ g σ f σ − ∇ g f | g σ ≤ C r σ | h | g σ ≤ C r − σ , (2.16) | Rm( g σ )( Y, Z, ∇ g σ f σ , U ) | g σ ≤ C r − σ , (2.17) for any unit vector fields Y , Z , U with respect to g σ . In particular, r σ satisfies estimates (2.7) , (2.8) and (2.9) which are uniform in σ ∈ [1 , . Remark 2.5.
In particular, estimate (2.12) not only implies that the potential functions f and f are equivalent but also that the exponential weights e f and e f are uniformly comparable.Proof. We only prove (2.12) and (2.16), the other estimates can be proved along the samelines.In order to prove (2.12), recall by (2.5) that if ( ϕ t ) t> denotes the flow generated by −∇ g f /t such that ϕ t | t =1 = Id M and if f ( t ) := ϕ ∗ t f , ∂ t ( tf ) = t R g ( t ) for t >
0. Byinvoking Theorem 2.1 again and by integrating the previous differential equation from t = 0to t = 1, one gets on each end of M , (cid:12)(cid:12)(cid:12)(cid:12) f − r (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ t R g ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, since t R g ( t ) is bounded on M × (0 , g andthe potential function f so that the triangle inequality implies the expected estimate, i.e. | f − f | ≤ C on each end of M for some positive constant C .Regarding the proof of (2.16), the main observation is that | g σ − g | g ≤ | g − g | g ≤ C r − by Theorem 2.1 together with the fact that ∇ g f = ∇ g f . Indeed, |∇ g σ f σ − ∇ g f | g = | ( σ − ∇ g σ f + (2 − σ ) ∇ g σ f − ∇ g f | g = | ( σ −
1) ( ∇ g σ f − ∇ g f ) + (2 − σ ) ( ∇ g σ f − ∇ g f ) | g ≤ C ( | g σ − g | g + | g σ − g | g ) r ≤ C | h | g r ≤ C r − , where we have used the estimate |∇ g i f i | g i = O ( r ) for i = 1 , (cid:3) Proposition 2.6.
Under the above assumptions and notations, define ∆ f σ h := ∆ g σ h + ∇ g σ ∇ gσ f σ h .Then, the following system holds outside a sufficiently large compact set of M for all σ ∈ [1 , : ∆ f σ h = L B ( g σ ) + R [ h ] , (2.18) (cid:12)(cid:12)(cid:12)(cid:12) ∇ g σ r σ ∂ r σ B − B (cid:12)(cid:12)(cid:12)(cid:12) g σ ≤ C (cid:0) r − σ | h | g σ + r − σ |∇ g σ h | g σ (cid:1) , (2.19) where | R [ h ] | g σ ≤ C r − σ (cid:0) | h | g σ + r − σ |∇ g σ h | g σ (cid:1) , (2.20) for some positive constant C uniform in σ ∈ [1 , .Proof. We first prove this proposition for σ = 1, i.e. with respect to the soliton metric g andthe potential function f .Notice that the associated Ricci flows to the soliton metrics g and g are respectively g ( t ) = tψ ∗ t g and g ( t ) = tψ ∗ t g since the flows of ∇ g f and ∇ g f are identified by Theorem2.1 as explained above. Define accordingly X := ∇ g f = ∇ g f outside a sufficiently largecompact set of M .Therefore, outside a compact set of M , − g ) − Ric( g )) = ( g − g ) − L X g + L X g . Now, apply formula [(2.10), Lemma 2.3] to the Lie derivative L X h defined on ( M n , g ) toget: L X g − L X g = L X h = ∇ g X h + Sym( h ◦ ∇ g , f )= ∇ g X h + h + Sym( h ◦ Ric( g ))where Sym( h ◦∇ g , f )( Y, Z ) := h ( ∇ g Y ∇ g f , Z )+ h ( Y, ∇ g Z ∇ g f ) for vector fields Y, Z ∈ T M.
As a first conclusion, we get: − g ) − Ric( g )) = − ∇ g X h − Sym( h ◦ Ric( g )) . On the other hand, according to [[Kot17], Lemma 4], see [KL12] for a similar expression: − g ) − Ric( g )) = g − ∗ ∇ g , h − L B ( g ) + g − ∗ Rm( g ) ∗ h + g − ∗ g − ∗ ∇ g h ∗ ∇ g h = g − ∗ ∇ g , h − L B ( g ) + 2 Rm( g ) ∗ h − Sym( h ◦ Ric( g )) + R [ h ] , where (cid:0) g − ∗ ∇ g , h (cid:1) ij := g kl ∇ g , kl ( h ) ij and where R [ h ] satisfies pointwise (2.20). Observe asin [KL12] that: g kl ∇ g , kl h = g kl ∇ g , kl h + (cid:16) g kl − g kl (cid:17) ∇ g , kl h = ∆ g h + R [ h ] , where | R [ h ] | g ≤ C | Rm( g ) | g | h | g + C (cid:0) | Rm( g ) | g | h | g + |∇ g , h | g (cid:1) | h | g + C |∇ g h | g . Note that the decay established in Theorem 2.1 yields | R [ h ] | g ≤ C r − (cid:0) | h | g + r − |∇ g h | g (cid:1) . which shows (2.18) and (2.20).To prove [(2.19), Proposition 2.6], notice by the definition of the Bianchi gauge given in(2.11) that: B ( t ) := div g ( t ) ( g ( t ) − g ( t )) − g ( t ) (cid:16) ∇ g ( t ) tr g ( t ) ( g ( t ) − g ( t )) , · (cid:17) = ψ ∗ t B , which implies by the soliton equation (2.13), ∂ t B ( t ) (cid:12)(cid:12)(cid:12) t =1 = − L X B = − ∇ g X B + ∇ g B X = − ∇ g X B + 12 B + Ric( g )( B ) . (2.21) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 11
Along the same lines as in [Kot17], one gets by differentiating the righthand side of (2.3): ∂ t B ( t ) (cid:12)(cid:12)(cid:12) t =1 = div g ( − g )) − g (cid:0) ∇ g tr g ( − g )) , · (cid:1) + h ∗ ∇ g Ric( g ) + ∇ g h ∗ Ric( g )= − g − div g ) Ric( g ) + g ( ∇ g tr g Ric( g ) , · ) − g ( ∇ g tr g Ric( g ) , · ) + h ∗ ∇ g Ric( g ) + ∇ g h ∗ Ric( g )= h ∗ O ( r − ) + ∇ g h ∗ O ( r − ) . (2.22)This fact uses the crucial Bianchi identity div g Ric( g ) − g ( ∇ g tr g Ric( g ) , · ) = 0 togetherwith the quadratic decay on the curvature and its derivatives. The combination of (2.21) and(2.22) leads to the desired result.If one considers an approximate soliton metric g σ and its associated approximate potentialfunction f σ for σ ∈ [1 , | ∆ f σ h − ∆ f h | g σ ≤ C r − σ (cid:0) | h | g σ + r − σ |∇ g σ h | g σ (cid:1) . (2.23)The proof of (2.23) is in turn due to Lemma 2.4 together with the following linearization ofthe rough laplacian: | ∆ g σ h − ∆ g h | g ≤ C X i =0 |∇ g ,i ( g σ − g ) | g |∇ g , − i h | g ≤ C X i =0 |∇ g ,i h | g |∇ g , − i h | g , fo some positive constant C uniform in σ ∈ [1 , ∇ g σ ∇ gσ f σ h and ∇ g σ ∇ gσ f σ B . Indeed, if T is any tensor defined on M , then pointwise: (cid:12)(cid:12)(cid:12) ∇ g σ ∇ gσ f σ T − ∇ g ∇ g f T (cid:12)(cid:12)(cid:12) g ≤ C (cid:16) |∇ g T | g |∇ g σ f σ − ∇ g f | g + r |∇ g σ T − ∇ g T | g (cid:17) ≤ C (cid:16) |∇ g T | g |∇ g σ f σ − ∇ g f | g + r |∇ g ( g σ − g ) | g | T | g (cid:17) ≤ C (cid:16) r |∇ g T | g | h | g + r |∇ g h | g | T | g (cid:17) where we have crucially used (2.16) in the last line. (cid:3) We now consider ˆ h := Ψ µ h . We recall from [Ber17, p. 17] that L m Ψ µ = 12 ( µ + n + m )Ψ µ + O µ,m ( r − Ψ µ ) , where O µ,m means that the estimate depends on µ and m .Combining this with (2.18), one can compute that (cid:18) L − µ + µ − n (cid:19) ˆ h = Ψ µ L B ( g ) + O µ,n ( r − )ˆ h + Ψ µ R [ h ] . The same computation is true for the operator L − µ defined with respect to the one-parameterfamilies of metrics ( g σ ) σ ∈ [1 , and potential functions ( f σ ) σ ∈ [1 , . In order to keep the notationsto a minimum, we denote such operator by the same symbol in the sequel.Choosing µ = n and recalling the properties of the quadratic term R [ h ] from Proposition2.6 together with the decay on h and its derivatives given by Theorem 2.1, we summarize thisdiscussion as follows: Corollary 2.7.
Under the same assumptions and notations as those of Proposition 2.6, thefollowing system holds outside a sufficiently large compact set of M for all σ ∈ [1 , : L − n ˆ h = Ψ n L B ( g σ ) + R [ˆ h ] , (2.24) (cid:12)(cid:12)(cid:12)(cid:12) ∇ g σ r σ ∂ r σ B − B (cid:12)(cid:12)(cid:12)(cid:12) g σ ≤ C (cid:0) r − σ | h | g σ + r − σ |∇ g σ h | g σ (cid:1) , where, | R [ˆ h ] | g σ ≤ C r − σ (cid:16) | ˆ h | g σ + |∇ g σ ˆ h | g σ (cid:17) . In particular, Corollary 2.7 shows that we have reduced our setting to that of [Ber17, Sec-tion 3] up to the Lie derivative term on the righthand side of (2.24).Because of Lemma 2.4, Proposition 2.6 and Corollary 2.7, we omit the reference to theparameter σ ∈ [1 , from now on unless we explicitly make reference to it .2.4. Frequency functions. On E R := { p | r ( p ) > R } , we consider C k ( ¯ E R ) with the (incomplete) norms k T k m = k T k m, = ˆ ¯ E R | T | Φ m and k T k m, = k T k m, + k∇ T k m, , together with the classes of tensors of controlled growth C km ( ¯ E R ) := { T ∈ C k ( ¯ E R ) | k T k m < ∞} and C km, ( ¯ E R ) := { T ∈ C k ( ¯ E R ) | k T k m, < ∞} . Fixing
R > R Σ and denote for ρ ≥ RB ( ρ ) := ˆ S ρ | ˆ h | |∇ r | and ˆ B ( ρ ) := Φ − n ( ρ ) B ( ρ ) . We also define the boundary flux F ( ρ ) := ˆ S ρ h ˆ h, ∇ − N ˆ h i = − ˆ S ρ h ˆ h, ∇ ∂ r ˆ h i|∇ r | and ˆ F ( ρ ) = Φ − n ( ρ ) F ( ρ ) , and the corresponding frequency functions N ( ρ ) := ρF ( ρ ) B ( ρ ) = ρ ˆ F ( ρ )ˆ B ( ρ ) . For ˆ h ∈ C − n, ( ¯ E R ) we letˆ D ( ρ ) := ˆ ¯ E ρ |∇ ˆ h | Φ − n and D ( ρ ) := Φ − n ( ρ ) − ˆ D ( ρ ) , be the Φ − n -weighted Dirichlet energy and a normalized version of it andˆ N ( ρ ) = ρ ˆ D ( ρ )ˆ B ( ρ ) = ρD ( ρ ) B ( ρ ) , the corresponding frequency functions. We also setˆ L ( ρ ) = ˆ E ρ h ˆ h, L − n ˆ h i Φ − n and L ( ρ ) = Φ − n ( ρ ) − ˆ L ( ρ ) , RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 13 so that if L − n h ∈ C − n ( ¯ E R ) then integration by parts is justified and givesˆ F ( ρ ) = ˆ D ( ρ ) + ˆ L ( ρ ) and F ( ρ ) = D ( ρ ) + L ( ρ ) . (2.25)We quote the two following lemmata whose proofs are verbatim those given in [Ber17]. Lemma 2.8. ([Ber17, Lemma 3.2])
We have B ′ ( ρ ) = n − ρ B ( ρ ) − F ( ρ ) + B ( ρ ) O ( ρ − ) and ˆ B ′ ( ρ ) = − n + 1 ρ ˆ B ( ρ ) − ρ B ( ρ ) − F ( ρ ) + ˆ B ( ρ ) O ( ρ − ) . Lemma 2.9. ([Ber17, Lemma 3.4])
There is an
R > so that for any ρ ≥ R , if ˆ h ∈ C ( ¯ A ρ +2 ρ ) satisfies − ≤ N ( ρ ′ ) ≤ N + for ρ ′ ∈ [ ρ, ρ + 2] and N + ≥ , then for all τ ∈ [0 , , (cid:16) − N + τρ (cid:17) B ( ρ ) ≤ B ( ρ + τ ) ≤ (cid:16) n + 3) τρ (cid:17) B ( ρ ) . Integral decay estimates
From now on, we consider an asymptotically conical expanding gradient Ricci soliton( M n , g, ∇ g f ) gauged outside a compact set as in Theorem 2.1. We consider a couple ( h, B )satisfying (2.18) and (2.19) and derive a priori integral bounds. For the sake of clarity, weomit the dependence of the Levi-Civita connection on the background metric g .Given a symmetric 2-tensor h ∈ H loc , r > r and m ∈ R , we define the following weightedenergy: ˇ D m ( h, r , r ) := ˆ A r ,r |∇ h | Ψ m . Similarly as in Section 2.4 for
R > R Σ and ρ ≥ R we defineˇ B ( ρ ) := ˆ S ρ | h | |∇ r | , as well as the boundary fluxˇ F ( ρ ) := ˆ S ρ h h, ∇ − N h i = − ˆ S ρ h h, ∇ ∂ r h i|∇ r | . (3.1) Remark 3.1.
Note that the proof of [Ber17, Lemma 3.2] depends only on the first variationformula and the weak conical hypotheses, so as in Lemma 2.8 we have ˇ B ′ ( ρ ) = n − ρ ˇ B ( ρ ) − F ( ρ ) + ˇ B ( ρ ) O ( ρ − ) . Lemma 3.2. ([Ber17, Lemma 9.2])
Let α ∈ R . Then there is a radius R such that for any r ≥ r ≥ R and h ∈ C ( A r ,r ) , ˆ S r r − α − |∇ r || h | Ψ m + ˆ A r ,r r − α | h | Ψ m ≤ C α r α +21 ˇ D m ( h, r , r ) + C α ˆ S r r − α − |∇ r || h | Ψ m . Proof.
The proof consists in applying the divergence theorem to the vector field V := r − α − | h | Ψ m ∂ r . Indeed, by (2.8) and (2.9),div V = n + m − α − r α +2 | h | Ψ m + 2 r α +1 h h, ∇ ∂ r h i Ψ m + 12 r α | h | Ψ m + | h | Ψ m O ( r − − α ) . The use of Young’s inequality then leads to the desired estimate. (cid:3)
The following lemma gives a priori integral estimates on the Bianchi gauge both withrespect to polynomial and exponential weights:
Lemma 3.3.
Given m ∈ R , there is a radius R = R ( n, m ) such that if r ≥ R and ( h, B ) satisfies (2.18) – (2.20) , then for all r > r ≥ R and ε ∈ (0 , , ˆ A r ,r (cid:16) (1 − ε ) r n + m + 22 + O ( r − ) (cid:17) |B| Ψ m + 12 ˆ S r r |∇ r ||B| Ψ m ≤ ˆ S r r |∇ r ||B| Ψ m + C ε ˆ A r ,r (cid:0) r − | h | + r − |∇ h | (cid:1) Ψ m , and, ˆ A r ,r (cid:16) n − α O ( r − ) (cid:17) |B| r − α + 12 ˆ S r |∇ r ||B| r − α ≤ ˆ S r |∇ r ||B| r − α + C ˆ A r ,r (cid:0) r − | h | + r − |∇ h | (cid:1) r − α . (3.2) Proof.
By integration by parts: ˆ A r ,r (cid:28) ∇ (cid:18) r (cid:19) , ∇|B| (cid:29) Ψ m = − ˆ A r ,r div (cid:18) Ψ m ∇ (cid:18) r (cid:19)(cid:19) |B| + 12 ˆ S r r |B| h ∂ r , N i Ψ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r r = − ˆ A r ,r (cid:18)(cid:28) ∇ (cid:18) r (cid:19) , ∇ (cid:18) r m ln r (cid:19)(cid:29) + ∆ r (cid:19) |B| Ψ m + 12 ˆ S r r |B| h ∂ r , N i Ψ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r r . Using (2.19) now leads to: ˆ A r ,r r n + m + 22 + O ( r − ) ! |B| Ψ m + 12 ˆ S r r |∇ r ||B| Ψ m ≤ ˆ S r r |∇ r ||B| Ψ m + C ˆ A r ,r (cid:0) r − | h | + r − |∇ h | (cid:1) |B| Ψ m ≤ ˆ S r r |∇ r ||B| Ψ m + C ε ˆ A r ,r (cid:0) r − | h | + r − |∇ h | (cid:1) Ψ m + ε ˆ A r ,r r |B| Ψ m , RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 15 where we have used Young’s inequality in the last line. This shows the desired inequality(3.3).As for the inequality (3.2) with polynomial weight, we proceed similarly by applying thedivergence theorem to the vector field V := |B| r − α ∇ f . (cid:3) Lemma 3.4. ([Ber17, Lemma 9.3])
Given m ∈ R and ε ∈ (0 , , there is a radius R = R ( n, m, ε ) such that if ( h, B ) satisfies (2.18) – (2.20) on E R , then for all r > r ≥ R , Ψ m ( r ) ˇ F ( r ) − Ψ m ( r ) ˇ F ( r ) ≥ − ε Ψ m ( r ) r ˇ B ( r ) − C ε ˆ S r r |B| Ψ m . Proof.
Using the definition of the flux F given in (3.1), integration by parts shows that:Ψ m ( r ) ˇ F ( r ) − Ψ m ( r ) ˇ F ( r ) = ˆ A r ,r |∇ h | Ψ m + ˆ A r ,r h h, L + m h i Ψ m = ˇ D m ( h, r , r ) + ˆ A r ,r D h, L B ( g ) + R [ h ] + m r ∇ ∂ r h E Ψ m = ˇ D m ( h, r , r ) + ˆ A r ,r h h, R [ h ] i Ψ m − h div(Ψ m h ) , Bi + 2 " ˆ S r h ( N , B )Ψ m r r + ˆ A r ,r m r h h, ∇ ∂ r h i Ψ m . Now, observe thanks to [(2.20), Proposition 2.6] that for ε > r ≥ R sufficiently large, | div(Ψ m h ) ||B| ≤ ε (cid:18) | h | + |∇ h | r (cid:19) Ψ m + C ε r |B| Ψ m , |h h, R [ h ] i| Ψ m ≤ C r (cid:0) | h | + r − | h ||∇ h | (cid:1) Ψ m ≤ ε (cid:18) | h | + |∇ h | r (cid:19) Ψ m . In particular, we get:Ψ m ( r ) ˇ F ( r ) − Ψ m ( r ) ˇ F ( r ) ≥ ˇ D m ( h, r , r ) − ε ˆ A r ,r (cid:18) | h | + |∇ h | r (cid:19) Ψ m − C ε ˆ A r ,r r |B| Ψ m − ˆ S r | h ||B| Ψ m − ˆ S r | h ||B| Ψ m − C ˆ A r ,r | h | + |∇ h | r Ψ m . According to Lemma 3.2 with α = 1, we get in case r ≥ R is sufficiently large,Ψ m ( r ) ˇ F ( r ) − Ψ m ( r ) ˇ F ( r ) ≥ (cid:18) − Cr − εr − Cr (cid:19) ˇ D m ( h, r , r ) − ε ˆ A r ,r | h | Ψ m − C ε ˆ A r ,r r |B| Ψ m − ˆ S r | h ||B| Ψ m − ˆ S r | h ||B| Ψ m − Cr Ψ m ( r ) ˆ S r |∇ r || h | ≥ (cid:18) − Cr (cid:19) ˇ D m ( h, r , r ) − ε ˆ A r ,r | h | Ψ m − C ε ˆ S r r |B| Ψ m + ˆ A r ,r (cid:0) r − | h | + r − |∇ h | (cid:1) Ψ m ! − ˆ S r (cid:0) ε r − | h | + C ε r |B| (cid:1) Ψ m − ˆ S r (cid:0) ε r − | h | + C ε r |B| (cid:1) Ψ m − Cr Ψ m ( r ) ˆ S r |∇ r || h | ≥ (cid:18) − Cr (cid:19) ˇ D m ( h, r , r ) − ε ˆ A r ,r | h | Ψ m − C ε ˆ S r r |B| Ψ m − ε ˆ S r r − | h | Ψ m − Cεr Ψ m ( r ) ˇ B ( r ) . where we have used Lemma 3.3 in the second and third inequalities. Using Lemma 3.2 with α = 0, we finally establish:Ψ m ( r ) ˇ F ( r ) − Ψ m ( r ) ˇ F ( r ) ≥ (cid:18) − Cr − Cr (cid:19) ˇ D m ( h, r , r ) − C ε ˆ S r r |B| Ψ m − Cεr Ψ m ( r ) ˇ B ( r ) ≥ − C ε ˆ S r r |B| Ψ m − Cεr Ψ m ( r ) ˇ B ( r ) , if r ≥ R is sufficiently large. (cid:3) The next lemma derives a differential inequality satisfied by B B ( ρ ) := ˆ S ρ |B| |∇ r | , ρ ≥ R, defined for R large enough. Lemma 3.5.
There exists a radius
R > such that for ρ ≥ R , B ′B ( ρ ) = n + 1 ρ B B ( ρ ) + O ( ρ − ) B B ( ρ ) + (cid:18) ˆ S ρ O ( r − ) | h | + O ( r − ) |∇ h | (cid:19) B B ( ρ )= n + 1 ρ B B ( ρ ) + O ( ρ − ) B B ( ρ ) + O ( ρ − ) ˇ B ( ρ ) + O ( ρ − ) ˆ S ρ |∇ h | . Proof.
According to the first variation formula and (2.19), B ′B ( ρ ) = ˆ S ρ |B| |∇ r | ( r − h X , N i H S ρ + 2 h r − ∇ X B , Bi|∇ r | + |B| r − X · |∇ r | = n − ρ B B ( ρ ) + 2 ˆ S ρ r − |B| |∇ r | + ˆ S ρ (cid:0) O ( r − ) | h | + O ( r − ) |∇ h | (cid:1) |B| + ˆ S ρ O ( r − ) |B| = n + 1 ρ B B ( ρ ) + O ( ρ − ) B B ( ρ ) + (cid:18) ˆ S ρ O ( r − ) | h | + O ( r − ) |∇ h | (cid:19) B B ( ρ ) . RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 17
This proves the expected estimate by the Cauchy-Schwarz inequality. (cid:3)
The main result of this section is the following exponential integral decay of a couple ( h, B )that satisfies (2.18)–(2.20) under the sole assumption of an a priori polynomial decay. Theorem 3.6.
Let ( h, B ) satisfy (2.18) – (2.20) . Assume S ρ (cid:0) | h | + ρ |B| (cid:1) = O ( ρ − ε ) , for some ε > . Then for all m ∈ R , there exists R > and β > such that for all ρ ≥ R , ˆ S ρ | h | ≤ C Φ m ( ρ ) , ρ ˆ S ρ |B| ≤ C Φ m ( ρ ) + Cρ n − e − βρ ˆ A R,ρ |∇ h | e β r r − n +1 , for some positive constant C = C ( n, m, h ) . Remark 3.7.
Ideally one would impose ffl S ρ | h | + ρ |B| = o (1) only to reach the same conclu-sions of Theorem 3.6. Nonetheless, assumption (3.6) will be satisfied in our main applications. Before proving Theorem 3.6, we ensure the integrability of the energy of a solution ( h, B )satisfying (2.18)–(2.20) in polynomial weighted L spaces: Lemma 3.8.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) . Assume ˆ E R (cid:0) | h | + |B| (cid:1) r − α < + ∞ , for some α ∈ R . Then, ˆ E R |∇ h | r − α < + ∞ . (3.3) Moreover, for γ ∈ (0 , , α ∈ R , there exists R such that if ρ ≥ R , ˆ S ρ r − α h h, ∇ N h i + (1 − γ ) ˆ E ρ |∇ h | r − α + n − α ˆ E ρ | h | r − α ≤ C γ ˆ E ρ | h | r − α − + C γ ˆ E ρ |B| r − α + Cρ − α B B ( ρ ) . (3.4) Proof.
To do so, let ρ ≥ ρ ≥ R and let η ρ ,ρ be a smooth cut-off function such that0 ≤ η ρ ,ρ ≤ η ρ ,ρ ≡ A ρ ,ρ , supp η ρ ,ρ ⊂ A ρ , ρ and ρ k |∇ k η ρ ,ρ | ≤ C on A ρ, ρ for k = 1 ,
2. Observe by the divergence theorem applied to the vector field V := r − α (cid:0) | h | ∇ η ρ ,ρ − η ρ ,ρ ∇| h | (cid:1) , that one gets:2 ˆ S ρ r − α h h, ∇ N h i = ˆ E ρ div (cid:0) r − α (cid:0) | h | ∇ η ρ ,ρ − η ρ ,ρ ∇| h | (cid:1)(cid:1) = ˆ E ρ r − α | h | ∆ η ρ ,ρ − r − α η ρ ,ρ ∆ | h | + h∇ r − α , ∇ η ρ ,ρ i| h | − h∇ r − α , ∇| h | i η ρ ,ρ . In particular, by Young’s inequality, for γ ∈ (0 , ˆ S ρ r − α h h, ∇ N h i + (1 − γ ) ˆ E ρ η ρ ,ρ |∇ h | r − α ≤ C γ ˆ E ρ r − α − | h | − ˆ E ρ h ∆ h, h i η ρ ,ρ r − α , (3.5)for some positive constant C independent of ρ that may vary from line to line. Here we haveused the assumption on the integrability of | h | r − α . Now, by using (2.18) and integration byparts, − ˆ E ρ h ∆ h, h i η ρ ,ρ r − α = 12 ˆ E ρ h∇ f, ∇| h | i η ρ ,ρ r − α + ˆ E ρ h R [ h ] , h i η ρ ,ρ r − α + 2 ˆ E ρ h div (cid:0) r − α η ρ ,ρ h ) , B (cid:11) + 2 ˆ S ρ r − α h ( B , N ) η ρ ,ρ = − ˆ E ρ div (cid:0) r − α η ρ ,ρ ∇ f (cid:1) | h | + ˆ E ρ h R [ h ] , h i η ρ ,ρ r − α + 2 ˆ E ρ h div (cid:0) r − α η ρ ,ρ h ) , B (cid:11) + ˆ S ρ (cid:16) h ( B , N ) − h∇ f, N i| h | (cid:17) η ρ ,ρ r − α . Thanks to [(2.20), Proposition 2.6], observe that: ˆ E ρ |h R [ h ] , h i| η ρ ,ρ r − α ≤ Cρ − ˆ E ρ |∇ h | η ρ ,ρ r − α + C ˆ E ρ | h | η ρ ,ρ r − α − , − ˆ E ρ div (cid:0) r − α η ρ ,ρ ∇ f (cid:1) | h | = ˆ E ρ (cid:18) α − n O ( r − ) (cid:19) η ρ ,ρ | h | r − α − ˆ E ρ h∇ η ρ ,ρ , ∇ f i| h | r − α , ˆ E ρ h div (cid:0) r − α η ρ ,ρ h ) , B (cid:11) ≤ γ ˆ E ρ (cid:0) |∇ h | + r − | h | (cid:1) η ρ ,ρ r − α + C γ ˆ E ρ |B| η ρ ,ρ r − α + C ˆ E ρ | h | |∇ η ρ ,ρ | r − α , (3.6)for any γ > ˆ S ρ (cid:16) h ( B , N ) − h∇ f, N i| h | (cid:17) η ρ ,ρ r − α ≤ Cρ − α B B ( ρ ) . (3.7) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 19
In particular, if γ is chosen sufficiently small and R is sufficiently large, injecting the previousestimates (3.6) and (3.7) back to (3.5) leads to: ˆ S ρ r − α h h, ∇ N h i + (1 − γ ) ˆ E ρ η ρ ,ρ |∇ h | r − α + n − α ˆ E ρ η ρ ,ρ | h | r − α ≤ C ˆ E ρ | h | (cid:0) r |∇ η ρ ,ρ | (cid:1) r − α − + C γ ˆ E ρ |B| η ρ ,ρ r − α + C ˆ A ρ, ρ | h | r − α + Cρ − α B B ( ρ ) ≤ C ˆ A ρ, ρ | h | r − α + C γ ˆ E ρ | h | r − α − + C γ ˆ E ρ |B| η ρ ,ρ r − α + Cρ − α B B ( ρ ) . (3.8)Here we have used that ∇ η ρ ,ρ = O ( r − ) and the fact that the integral on A ρ, ρ is boundedindependently of ρ ≥ ρ by assumption. This ends the proof of (3.3).In order to prove (3.4), we let ρ go to + ∞ in (3.8). (cid:3) Proof of Theorem 3.6.
On the one hand, Remark 3.1 and Lemma 3.4 applied to R ≤ r ≤ r := ρ gives for ε > B ′ ( ρ ) = n − ρ ˇ B ( ρ ) − F ( ρ ) + O ( ρ − ) ˇ B ( ρ )= n − ρ ˇ B ( ρ ) − − m ( ρ )Ψ m ( r ) ˇ F ( r ) + O ( ρ − ) ˇ B ( ρ )+ 2Φ − m ( ρ ) (cid:0) Ψ m ( r ) ˇ F ( r ) − Ψ m ( ρ ) ˇ F ( ρ ) (cid:1) ≥ n − ρ ˇ B ( ρ ) − − m ( ρ )Ψ m ( r ) ˇ F ( r ) + O ( ρ − ) ˇ B ( ρ ) − ε Φ − m ( ρ )Ψ m ( ρ ) ρ ˇ B ( ρ ) − C ε Φ − m ( ρ ) ˆ S ρ r |B| Ψ m = n − − ερ ˇ B ( ρ ) − − m ( ρ )Ψ m ( r ) ˇ F ( r ) + O ( ρ − ) ˇ B ( ρ ) − C ε ˆ S ρ r |B| . In particular, there is a positive constant A such that: e − A ρ − ˇ Bρ n − − ε ! ′ ( ρ ) ≥ A ρ e − A ρ − ˇ B ( ρ ) ρ n − − ε − C ( h, r )Φ − m − ( n − − ε ) ( ρ ) − C ε ρ n − − ε B B ( ρ ) . (3.9) On the other hand, observe that Lemma 3.5 ensures that (cid:18) ρ B B ρ n − − ε (cid:19) ′ ( ρ ) = n + 1 − ( n − − ε ) ρ (cid:18) ρ B B ( ρ ) ρ n − − ε (cid:19) + O ( ρ − ) (cid:18) ρ B B ( ρ ) ρ n − − ε (cid:19) + O ( ρ − ) ˇ B ( ρ ) ρ n − − ε + O ( ρ − ) ˆ S ρ r − ( n − − ε ) |∇ h | = (4 + ε ) B B ( ρ ) ρ n − − ε + O ( ρ − ) (cid:18) ρ B B ( ρ ) ρ n − − ε (cid:19) + O ( ρ − ) ˇ B ( ρ ) ρ n − − ε + O ( ρ − ) ˆ S ρ r − ( n − − ε ) |∇ h | . This in turn implies that there is a positive constant A such that: e − A ρ − ρ B B ρ n − − ε ! ′ ( ρ ) ≥ e − A ρ − (4 + ε ) B B ( ρ ) ρ n − − ε − Cρ − ˇ B ( ρ ) ρ n − − ε − Cρ ˆ S ρ r − ( n − − ε ) |∇ h | . (3.10)Therefore, combining (3.9) with (3.10) gives for any A > e − A ρ − ˇ Bρ n − − ε + A e − A ρ − ρ B B ρ n − − ε ! ′ ( ρ ) ≥ − C ( h, r )Φ − m − ( n − − ε ) ( ρ ) − A Cρ ˆ S ρ r − ( n − − ε ) |∇ h | + (cid:16) A e − A ρ − (4 + ε ) − C ε (cid:17) B B ( ρ ) ρ n − − ε + (cid:16) A e − A ρ − ρ − − A Cρ − (cid:17) ˇ B ( ρ ) ρ n − − ε . Now choose R such that if ρ ≥ R then e − A ρ − ≥ and A so large that A ≥ C ε . Increasing A so that A e − A ρ − ρ − ≥ A Cρ − on this region implies then: (cid:18) e − A ρ − ˇ Bρ n − − ε + A e − A ρ − ρ B B ρ n − − ε (cid:19) ′ ( ρ ) ≥ − C ( h, r )Φ − m − ( n − − ε ) ( ρ ) − A Cρ ˆ S ρ r − ( n − − ε ) |∇ h | . (3.11)By assumption, there exists ε > ρ → + ∞ e − A ρ − ˇ B ( ρ ) ρ n − − ε + A e − A ρ − ρ B B ( ρ ) ρ n − − ε = 0 . RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 21
Therefore, by integrating (3.11) between ρ and + ∞ and using the co-area formula, one getsas an intermediate result: e − A ρ − ˇ B ( ρ ) ρ n − − ε + A e − A ρ − ρ B B ( ρ ) ρ n − − ε ≤ C ( h, r ) ˆ + ∞ ρ Φ − m − ( n − − ε ) ( s ) ds + A C ˆ E ρ r − ( n +1 − ε ) |∇ h | ≤ C ( h, r ) Φ − m − ( ρ ) ρ n − − ε + A C ˆ E ρ r − ( n +1 − ε ) |∇ h | . (3.12)Notice that the integral on the righthand side is well-defined thanks to Lemma 3.8 applied to α := n + 1 − ε since by assumption (3.6) and the co-area formula, ˆ E R (cid:0) | h | + |B| (cid:1) r − ( n +1 − ε ) = ˆ + ∞ R ˆ S ρ | h | + |B| |∇ r | ! ρ − ( n +1 − ε ) dρ ≤ C ˆ + ∞ R ρ − dρ < + ∞ . Here, we have not used the whole strength of the assumption on B .According to [(3.4), Lemma 3.8] applied to α := n + 1 − ε and γ = ,12 ˆ E ρ r − ( n +1 − ε ) |∇ h | ≤ − ρ n +1 − ε ˆ S ρ h h, ∇ N h i + ˆ E ρ (cid:18) − ε C r − (cid:19) r − ( n +1 − ε ) | h | + C ˆ E ρ r − ( n +1 − ε ) |B| + Cρ B B ( ρ ) ρ n − − ε . (3.13)Let us apply [(3.2), Lemma 3.3] to α := n + 1 − ε and r := ρ and r = + ∞ , legitimated byassumption (3.6), to get: ˆ E ρ r − ( n +1 − ε ) |B| ≤ C ˆ E ρ (cid:0) r − | h | + r − |∇ h | (cid:1) r − ( n +1 − ε ) . (3.14)Once inequality (3.14) is injected into (3.13), one can absorb the integral of |∇ h | on E ρ toget:14 ˆ E ρ r − ( n +1 − ε ) |∇ h | ≤ − ρ n +1 − ε ˆ S ρ h h, ∇ N h i + ˆ E ρ (cid:18) − ε C r − (cid:19) r − ( n +1 − ε ) | h | + Cρ B B ( ρ ) ρ n − − ε . (3.15)To conclude, let us plug (3.15) back to (3.12) to get for ρ ≥ R , R being sufficiently large:ˇ B ( ρ ) ρ n − − ε + A ρ B B ( ρ ) ρ n − − ε ≤ C ( h, r ) Φ − m − ( ρ ) ρ n − − ε − A Cρ n +1 − ε ˆ S ρ h h, ∇ N h i + A C ˆ E ρ (cid:18) − ε O ( r − ) (cid:19) r − ( n +1 − ε ) | h | . (3.16)If ε can be taken to be larger than 1 then one can simplify (3.16) one step further to get:ˇ B ( ρ ) ρ n − + A ρ B B ( ρ ) ρ n − ≤ C ( h, r ) Φ − m − ( ρ ) ρ n − − A Cρ n − ˆ S ρ h h, ∇ N h i . (3.17) Thanks to Remark 3.1, we observe that ddρ (cid:16) e Cρ − ρ − ( n − ˇ B (cid:17) = ddρ (cid:18) e Cρ − ˆ S ρ r − ( n − | h | |∇ r | (cid:19) ≤ e Cρ − ρ − ( n − ˆ S ρ h h, ∇ N h i , for some sufficiently large constant C and we conclude with the help of (3.17) that: ddρ (cid:16) e Cρ − ρ − ( n − ˇ B (cid:17) + ρ A C e Cρ − ρ − ( n − ˇ B ( ρ ) + C ρ B B ( ρ ) ρ n − ≤ C ( h, r ) Φ − m +1 ( ρ ) ρ n − , where C is a positive constant that may vary from line to line. Gr¨onwall’s inequality givesfor ρ ≥ r : ρ − ( n − ˇ B ( ρ ) + Ce − ρ A C ˆ ρρ e s A C s B B ( s ) s n − ds ≤ C ( h, r ) e − ρ A C + C ( h, r ) e − ρ A C ˆ ρr e s A C Φ − m − n +2 ( s ) ds. It can be shown by integration by parts that the right-hand side of the previous inequality isequivalent to Φ − m − n ( ρ ) as ρ tends to + ∞ , in particular, this shows that ˆ S ρ | h | |∇ r | ≤ C ( h, r )Φ − m − ( ρ ) , ρ ≥ r ≥ R, (3.18)which yields the first part of Theorem 3.6. We are also left with the following estimate onthe Bianchi gauge: e − ρ A C ˆ ρρ e s A C s B B ( s ) s n − ds ≤ C ( h, r ) e − ρ A C + C ( h, r )Φ − m − n ( ρ ) . (3.19) Claim 3.9.
For β > , there exist C > and ρ > such that if ρ > ρ , C − ρ B B ( ρ ) ρ n − ≤ e − βρ ˆ ρρ e βs s B B ( s ) s n − ds + Φ − m − n − ( ρ )+ e − βρ ˆ A ρ ,ρ |∇ h | e β r r − n +1 . Proof of Claim 3.9.
Let us integrate by parts as follows: ˆ ρρ e βs s B B ( s ) s n − ds = (cid:18) ˆ ρρ e βs ds (cid:19) ρ B B ( ρ ) ρ n − − ˆ ρρ (cid:18) ˆ sρ e βu du (cid:19) (cid:18) s B B s n − (cid:19) ′ ( s ) ds ≥ Ce βρ ρ B B ( ρ ) ρ n − − ˆ ρρ (cid:18) ˆ sρ e βu du (cid:19) (cid:18) s B B s n − (cid:19) ′ ( s ) ds, (3.20) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 23 where used that assumption that ρ ≥ ρ . Now, according to Lemma 3.5, ˆ ρρ (cid:18) ˆ sρ e βu du (cid:19) (cid:18) s B B s n − (cid:19) ′ ( s ) ds = ˆ ρρ (cid:18) ˆ sρ e βu du (cid:19) (cid:18) s + O ( s − ) (cid:19) s B B ( s ) s n − ds + ˆ ρρ (cid:18) ˆ sρ e βu du (cid:19) (cid:18) O ( s − ) ˇ B ( s ) s n − + O (1) s − ( n − ˆ S s |∇ h | (cid:19) ≤ Cρ ˆ ρρ e βs s B B ( s ) s n − ds + C ˆ ρρ e βs Φ − m − n − ( s ) ds + C ˆ ρρ e βs s − n +1 ˆ S s |∇ h | ds. (3.21)Here we have used (3.18) in the second inequality. Injecting (3.21) into (3.20) leads to theexpected claim once we invoke the co-area formula. (cid:3) Thanks to Claim 3.9 applied to β := A C , estimate (3.19) leads to the second part ofTheorem 3.6. (cid:3) Corollary 3.10.
Let ( h, B ) satisfy (2.18) – (2.20) . Assume S ρ (cid:0) | h | + ρ |B| (cid:1) = O ( ρ − ε ) , (3.22) for some ε > . Then, ˆ E R (cid:0) | h | + |∇ h | (cid:1) Ψ m < + ∞ , (3.23) for all m ∈ R . In particular, ˆ h ∈ C − n, ( E R ) , i.e. ˆ E R (cid:16) | ˆ h | + |∇ ˆ h | (cid:17) Φ − n < + ∞ . Proof.
According to Theorem 3.6 applied to − m − ˆ E R | h | Ψ m = ˆ + ∞ R ˆ S ρ | h | |∇ r | Ψ m ( ρ ) ≤ C ˆ + ∞ R Φ − m − ( ρ )Ψ m ( ρ ) dρ ≤ C ˆ + ∞ R ρ − dρ < + ∞ . (3.24)Following Bernstein [Ber17, Proof of Theorem 9 . η ρ ,ρ , ρ ≥ ρ ≥ R , be a radialsmooth cut-off function which is non-increasing on E ρ such that 0 ≤ η ρ ,ρ ≤ η ρ ,ρ ≡ A ρ ,ρ , supp η ρ ,ρ ⊂ A ρ , ρ and ρ k |∇ k η ρ ,ρ | ≤ C on A ρ, ρ for k = 1 ,
2. Let us integrate by parts as follows by using (2.18): ˆ E R ( L + m η ρ ,ρ ) | h | Ψ m = ˆ E R η ρ ,ρ L + − n | h | Ψ m = 2 ˆ E R η ρ ,ρ |∇ h | Ψ m + 2 ˆ E R η ρ ,ρ hL + m h, h i Ψ m = 2 ˆ E R η ρ ,ρ |∇ h | Ψ m + 2 ˆ E R η ρ ,ρ D L B ( g ) + m r ∇ ∂ r h + R [ h ] , h E Ψ m = 2 ˆ E R η ρ ,ρ |∇ h | Ψ m − ˆ E R h div (cid:0) Ψ m η ρ ,ρ h (cid:1) , Bi + ˆ E R η ρ ,ρ D m r ∇ ∂ r h + R [ h ] , h E Ψ m . Thanks to [(2.20), Proposition 2.6] and Young’s inequality, one gets:12 ˆ E R η ρ ,ρ |∇ h | Ψ m ≤ ˆ E R (cid:18) L + m η ρ ,ρ + (cid:18) m r + 1 (cid:19) η ρ ,ρ + |∇ η ρ ,ρ | (cid:19) | h | Ψ m + C ˆ E R r η ρ ,ρ |B| Ψ m . (3.25)We are left with estimating the last integral on the righthand side of the previous inequalityin terms of the energy of h with weight Ψ m . To do so, we invoke Theorem 3.6 applied to − m − ˆ E R r η ρ ,ρ |B| Ψ m ≤ ˆ + ∞ R η ρ ,ρ s B B ( s )Ψ m ( s ) ds ≤ C ˆ + ∞ R η ρ ,ρ Φ − m − ( s )Ψ m ( s ) ds + C ˆ + ∞ R η ρ ,ρ s n − e − βs ˆ A R,s |∇ h | e β r r − n +1 ! Ψ m ( s ) ds ≤ C ˆ + ∞ R s − ds + C ˆ E R (cid:18) ˆ + ∞ r s n − e − βs Ψ m ( s ) η ρ ,ρ ds (cid:19) |∇ h | e β r r − n +1 . Now, since η ρ ,ρ is decreasing on E ρ , one gets by a straightforward integration by parts thatfor r ≥ R : ˆ + ∞ r s n − e − βs Ψ m ( s ) η ρ ,ρ ds ≤ Cη ρ ,ρ ( r ) r n − e − β r Ψ m ( r ) + Cχ A ρ , ρ . Therefore, ˆ E R r η ρ ,ρ |B| Ψ m ≤ C ( h, ρ ) + C ˆ E R η ρ ,ρ |∇ h | r − Ψ m . (3.26)Pulling (3.26) into (3.25) leads to:12 ˆ E R η ρ ,ρ |∇ h | Ψ m ≤ ˆ E R (cid:18) L + m η ρ ,ρ + (cid:18) m r + 1 (cid:19) η ρ ,ρ + |∇ η ρ ,ρ | (cid:19) | h | Ψ m + C ( h, ρ )+ C ˆ E R η ρ ,ρ |∇ h | r − Ψ m RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 25 ≤ C ˆ E R | h | Ψ m + Cρ ˆ E R η ρ ,ρ |∇ h | Ψ m + C ( h, ρ ) . Here we have used (3.24) in the second inequality. This gives us the integrability of |∇ h | with respect to the weight Ψ m by choosing R large enough by absorption. (cid:3) Preliminary integral bounds
Recall that Corollary 3.10 ensures that ˆ h and B are in weighted L with respect to Φ m forall m ∈ R under the assumption that we recall here for the sake of clarity: S ρ (cid:0) | h | + ρ |B| (cid:1) = O ( ρ − ε ) , (4.1)for some ε >
1. We can then record the following result:
Proposition 4.1. ([Ber17, Proposition 3.1])
There is an
R > so that if ρ ≥ R and ˆ h ∈ C − n, ( ¯ E R ) then there exists a positive constant C such that: ˆ E ρ | ˆ h | Φ − n ≤ Cρ ˆ D ( ρ ) + Cρ ˆ B ( ρ ) . For later reference we note that integration by parts gives for all ρ ≥ ˆ ρ +1 ρ Φ − n ( t ) dt = 2 ρ Φ − n ( ρ ) − ρ + 1 Φ − n ( ρ + 1) − n + 1) ˆ ρ +1 ρ Φ − n − ( t ) dt = 2 ρ Φ − n ( ρ ) + Φ − n ( ρ ) O ( ρ − ) (4.2)The following lemma starts estimating the L -norm of the Bianchi one-form B on (super-)levelsets of r in terms of ˆ h if ( h, B ) ∈ C ( E R ) satisfies (2.18)–(2.20). Lemma 4.2. (Integral estimates on the Bianchi gauge) Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) as well as (4.1) . Then we have for ρ ≥ R , ˆ S ρ r |∇ r | |B| Ψ + ˆ E ρ r |B| Ψ ≤ C ˆ E ρ (cid:16) r − |∇ ˆ h | + r − | ˆ h | (cid:17) Φ − n ≤ Cρ − ˆ D ( ρ ) + Cρ − ˆ B ( ρ ) . (4.3) Proof.
By integration by parts, ˆ E ρ (cid:28) ∇ (cid:18) r (cid:19) , ∇|B| (cid:29) Ψ = ˆ E ρ (cid:28) ∇ (cid:18) e r (cid:19) , ∇|B| (cid:29) = − ˆ E ρ ∆ (cid:18) e r (cid:19) |B| − ˆ S ρ (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) |B| Ψ = − ˆ E ρ (cid:18) r n (cid:19) |B| Ψ − ˆ S ρ (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) |B| Ψ . (4.4) Now, thanks to (2.19), the lefthand side of (4.4) can be estimated as follows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ (cid:18)(cid:28) ∇ (cid:18) r (cid:19) , ∇|B| (cid:29) − |B| (cid:19) Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ E ρ (cid:16) r − |∇ ˆ h | + r − | ˆ h | (cid:17) |B| r n ≤ C ˆ E ρ (cid:16) r − |∇ ˆ h | + r − | ˆ h | (cid:17) Φ − n + 18 ˆ E ρ r |B| Ψ , (4.5)where we have used Young’s inequality in the last line.Combining (4.4) and (4.5) leads to the first part of the desired inequality (4.3): ˆ S ρ (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) |B| Ψ + ˆ E ρ (cid:18) r n (cid:19) |B| Ψ ≤ C ˆ E ρ (cid:16) r − |∇ ˆ h | + r − | ˆ h | (cid:17) Φ − n . The second inequality in (4.3) is obtained from the previous one together with Proposition4.1. (cid:3)
Lemma 4.3. ([Ber17, Lemma 3.3])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Thenthere exist C > and ρ ≥ R such that if ρ ≥ ρ , | ˆ L ( ρ ) | ≤ ρ − ˆ D ( ρ ) + Cρ − ˆ B ( ρ ) (4.6) ≤ ρ − ˆ F ( ρ ) + Cρ − ˆ B ( ρ ) . (4.7) In particular, if B ( ρ ) > , (cid:12)(cid:12)(cid:12) N ( ρ ) − ˆ N ( ρ ) (cid:12)(cid:12)(cid:12) ≤ ρ − N ( ρ ) + Cρ − . (4.8) Proof.
Let us integrate by parts by recalling the definition of ˆ h := Ψ n · h, ˆ L ( ρ ) = ˆ E ρ h ˆ h, L − n ˆ h i Φ − n = ˆ E ρ h ˆ h, Ψ n L B ( g ) + R [ˆ h ] i Φ − n = ˆ E ρ h ˆ h, L B ( g ) i r − n + ˆ E ρ h ˆ h, R [ˆ h ] i Φ − n = − ˆ E ρ hB , div (cid:16) r − n ˆ h (cid:17) i − ˆ S ρ ˆ h ( B , N ) r − n + ˆ E ρ h ˆ h, R [ˆ h ] i Φ − n . RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 27
In particular, for ε ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ˆ L ( ρ ) − ˆ E ρ h ˆ h, R [ˆ h ] i Φ − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ E ρ |B| (cid:12)(cid:12)(cid:12) div (cid:16) r − n ˆ h (cid:17)(cid:12)(cid:12)(cid:12) + 2 ˆ S ρ | ˆ h ||B| r − n ≤ C ˆ E ρ |B| (cid:16) |∇ ˆ h | + r − | ˆ h | (cid:17) r − n + 2 ˆ S ρ | ˆ h ||B| r − n ≤ C ε ˆ E ρ r |B| Ψ + ε ˆ E ρ r − (cid:16) |∇ ˆ h | + r − | ˆ h | (cid:17) Φ − n + C r − ˆ S ρ | ˆ h | Φ − n + ρ ˆ S ρ r |B| Ψ ≤ (cid:0) ε + Cρ − (cid:1) ˆ E ρ r − |∇ ˆ h | Φ − n + ( C ε + ε ) ˆ E ρ r − | ˆ h | Φ − n + C r − ˆ S ρ | ˆ h | Φ − n + Cρ ˆ E ρ (cid:16) r − |∇ ˆ h | + r − | ˆ h | (cid:17) Φ − n ≤ (cid:0) ε + Cρ − (cid:1) ˆ E ρ r − |∇ ˆ h | Φ − n + C ε ˆ E ρ r − | ˆ h | Φ − n + C r − ˆ S ρ | ˆ h | Φ − n , (4.9)where C ε is a positive constant that may vary from line to line. Here we have used Lemma4.2 in the fourth inequality and Young’s inequality in the third line.Finally, by Corollary 2.7 and Proposition 4.1, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h ˆ h, R [ˆ h ] i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ E ρ r − | ˆ h | (cid:16) | ˆ h | + |∇ ˆ h | (cid:17) Φ − n ≤ Cρ ˆ D ( ρ ) + Cρ ˆ B ( ρ ) + ˆ E ρ r − | ˆ h ||∇ ˆ h | Φ − n ≤ Cρ ˆ D ( ρ ) + Cρ ˆ B ( ρ ) + ˆ E ρ (cid:16) C ε r − | ˆ h | + ε r − |∇ ˆ h | (cid:17) Φ − n ≤ (cid:18) ε + Cρ (cid:19) ρ − ˆ D ( ρ ) + Cρ ˆ B ( ρ ) , (4.10)for any ε > C , C ε are positive constants that may vary from line to line.To conclude, we use the definitions of the various quantities defined at the end of Section2 together with the previous estimates (4.9) and (4.10): | ˆ L ( ρ ) | ≤ (cid:0) ε + Cρ − (cid:1) ρ − ˆ D ( ρ ) + C ε ρ − ˆ B ( ρ )+ C ε ρ − ˆ D ( ρ ) + C ε ρ − ˆ B ( ρ ) ≤ (cid:0) ε + C ε ρ − (cid:1) ρ − ˆ D ( ρ ) + C ε ρ − ˆ B ( ρ ) , where we have used Proposition 4.1 in the first line. This proves the expected result bychoosing ε small enough and by considering ρ ≥ ρ so large that ε + C ε ρ − ≤ .This establishes (4.6): inequality (4.7) is obtained thanks to identity (2.25) together with(4.6) and Young’s inequality. (cid:3) Corollary 4.4. ([Ber17, Lemma 3.5])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) .Then either h ≡ on E ρ for some ρ ≥ R or B ( ρ ) > for all ρ ≥ R .Proof. We follow Bernstein’s arguments very closely. Assume that B ( ρ ) = 0. Then h ≡ S ρ which implies that ˆ F ( ρ ) = 0. In particular, [(4.7), Lemma 4.3] ensures that ˆ L ( ρ ) = 0too. Thanks to identity (2.25), we then get that ˆ h is parallel on E ρ . As a consequence, thenorm | ˆ h | is constant on any connected component of E ρ and vanishes on S ρ , so h ≡ E ρ .Alternatively, one can use Proposition 4.1 to reach the same conclusion. (cid:3) Lemma 4.5. ([Ber17, Lemma 3.3])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρ ˆ E ρ |∇ N ˆ h | Φ − n ! (cid:18) ˆ D ( ρ ) + 1 ρ ˆ B ( ρ ) (cid:19) + Cρ ˆ D ( ρ ) + 1 ρ ˆ D ( ρ ) ˆ B ( ρ ) ! + C ˆ S ρ |∇ ˆ h | Φ − n ! ˆ S ρ r |B| Ψ ! . Proof.
Observe first that Corollary 2.7 implies: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h∇ X ˆ h, L B ( g ) i r − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ˆ E ρ |∇ X ˆ h || R [ˆ h ] | Φ − n . (4.11)Let us handle the non-linear terms first as follows: ˆ E ρ |∇ X ˆ h || R [ˆ h ] | Φ − n ≤ ˆ E ρ |∇ N ˆ h | Φ − n ! ˆ E ρ r | R [ˆ h ] | Φ − n ! ≤ C ˆ E ρ |∇ N ˆ h | Φ − n ! ˆ E ρ r − (cid:16) | ˆ h | + |∇ ˆ h | (cid:17) Φ − n ! ≤ Cρ ˆ E ρ |∇ N ˆ h | Φ − n ! (cid:18) ˆ D ( ρ ) + 1 ρ ˆ B ( ρ ) (cid:19) . Here we have used Cauchy-Schwarz inequality in the first line together with Corollary 2.7 inthe second line and Proposition 4.1 in the last line.To handle the first integral term on the righthand side of (4.11), we first establish thefollowing estimate:12 h∇ X ˆ h, L B ( g ) i r − n = div (cid:16)(cid:16) ∇ X ˆ h ( B ) − h div ˆ h, Bi X (cid:17) r − n (cid:17) + n r − n − ∇ X ˆ h ( B , ∇ r )+ O (cid:16) r − |B| (cid:16) r − | ˆ h | + |∇ ˆ h | (cid:17) + r − (cid:16) | ˆ h | + r − |∇ ˆ h | (cid:17) |∇ ˆ h | Φ − n (cid:17) . (4.12) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 29
To prove (4.12) we recall that X = r ∇ g r |∇ g r | and g ( ∇ Y X , Z ) = g ( Y, Z ) + O ( r − | Y || Z | ) , (4.13)for any tangent vectors Y, Z from the beginning of Section 2. We compute h∇ X ˆ h, ∇Bi r − n = X l ∇ l ˆ h ij ∇ i B j r − n = ∇ i (cid:16) X l ∇ l ˆ h ij B j r − n (cid:17) − ∇ i ˆ h ij B j r − n − X l ∇ i ∇ l ˆ h ij B j r − n + n X l ∇ l ˆ h ij B j r − n − ∇ i r + O ( r − n − |∇ ˆ h ||B| )= ∇ i (cid:16) X l ∇ l ˆ h ij B j r − n (cid:17) − ∇ i ˆ h ij B j r − n − X l ∇ l ∇ i ˆ h ij B j r − n + X l (cid:16) Rm( g ) pili ˆ h pj + Rm( g ) pilj ˆ h ip (cid:17) B j r − n + n X l ∇ l ˆ h ij B j r − n − ∇ i r + O ( r − n − |∇ ˆ h ||B| )= ∇ i (cid:16) X l ∇ l ˆ h ij B j r − n (cid:17) − ∇ l (cid:16) X l ∇ i ˆ h ij B j r − n (cid:17) − ∇ i ˆ h ij B j r − n − h Ric( g )( X ) , ˆ h ( B ) i g r − n + Rm( g ) pi X B ˆ h ip r − n + n ∇ X ˆ h ij B j r − n − ∇ i r + div( X ) ∇ i ˆ h ij B j r − n + ∇ i ˆ h ij ∇ X B j r − n − n ∇ i ˆ h ij B j r − n + O ( r − n − |∇ ˆ h ||B| + r − n − | ˆ h ||B| )= ∇ i (cid:16) X l ∇ l ˆ h ij B j r − n (cid:17) − ∇ l (cid:16) X l ∇ i ˆ h ij B j r − n (cid:17) + (div( X ) − n −
1) div ˆ h ( B ) r − n + div ˆ h ( ∇ X B ) r − n + n ∇ X ˆ h ( B , ∇ r ) r − n − + O ( r − n − |∇ ˆ h ||B| + r − n − | ˆ h ||B| ) . (4.14)Here we have used commutation formulas in the third line together with identities (2.15) and(2.17) to ensure that Rm( g )( X , · , · , · ) = O ( r − ) in the fifth equality.Now note that (4.13) implies thatdiv( X ) = n + O ( r − )as well as together with (2.8) and (2.19) that ∇ X B = 1 |∇ r | g ∇ ∇ f B = 1 |∇ r | g (cid:0) B + O ( r − ) | h | + O ( r − ) |∇ h | ) (cid:1) = B + O ( r − ) B + O ( r − ) | h | + O ( r − ) |∇ h | . Combining this with (4.14) we see that h∇ X ˆ h, ∇Bi r − n = ∇ i (cid:16) X l ∇ l ˆ h ij B j r − n (cid:17) − ∇ l (cid:16) X l ∇ i ˆ h ij B j r − n (cid:17) + n ∇ X ˆ h ij B j r − n − ∇ i r + O ( r − n − |∇ ˆ h ||B| + r − n − | ˆ h ||B| + r − n − |∇ ˆ h || h | + r − n − |∇ ˆ h ||∇ h | )= div (cid:16)(cid:16) ∇ X ˆ h ( B ) − h div ˆ h, Bi g X (cid:17) r − n (cid:17) + n r − n − ∇ X ˆ h ( B , ∇ r ) + O ( r − n − |∇ ˆ h ||B| + r − n − | ˆ h ||B| + r − n − |∇ ˆ h || h | + r − n − |∇ ˆ h ||∇ h | )= div (cid:16)(cid:16) ∇ X ˆ h ( B ) − h div ˆ h, Bi g X (cid:17) r − n (cid:17) + n r − n − ∇ X ˆ h ( B , ∇ r )+ O (cid:16) r − |B| (cid:16) r − | ˆ h | + |∇ ˆ h | (cid:17) + r − (cid:0) | h | + r − |∇ h | (cid:1) |∇ ˆ h | (cid:17) r − n . Recall that ˆ h := Ψ n · h . In particular, (cid:0) | h | + r − |∇ h | (cid:1) r − n = O (cid:16) | ˆ h | + r − |∇ ˆ h | (cid:17) Φ − n . Thisestablishes (4.12).Now, recalling (2.24), the divergence theorem applied to (4.12) leads to: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S ρ |∇ ˆ h ||B| r − n + C ˆ E ρ |∇ N ˆ h ||B| r − n + Cρ ˆ E ρ ( r − | ˆ h | + |∇ ˆ h | ) |B| r − n + C ˆ E ρ r − (cid:16) | ˆ h | + r − |∇ ˆ h | (cid:17) |∇ ˆ h | Φ − n ≤ C ˆ S ρ |∇ ˆ h | Φ − n ! ˆ S ρ r |B| Ψ ! + Cρ ˆ E ρ |∇ N ˆ h | Φ − n ! ˆ E ρ r |B| Ψ ! + Cρ ˆ E ρ r − ( r − | ˆ h | + |∇ ˆ h | ) Φ − n ! ˆ E ρ r |B| Ψ ! + Cρ ˆ E ρ |∇ ˆ h | Φ − n + Cρ ˆ E ρ | ˆ h | Φ − n ! ˆ E ρ |∇ ˆ h | Φ − n ! ≤ C ˆ S ρ |∇ ˆ h | Φ − n ! ˆ S ρ r |B| Ψ ! + Cρ ˆ E ρ |∇ N ˆ h | Φ − n ! (cid:18) ˆ D ( ρ ) + 1 ρ ˆ B ( ρ ) (cid:19) + Cρ ˆ D ( ρ ) + 1 ρ ˆ D ( ρ ) ˆ B ( ρ ) ! Here we have used Proposition 4.1 and Lemma 4.2 in the third inequality. (cid:3) Frequency bounds
We adapt [Ber17, Proposition 4 .
2] and [Ber17, Corollary 4 .
3] that hold true in general:
RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 31
Proposition 5.1.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then if ρ ≥ R , ˆ D ′ ( ρ ) = − ρ ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n − ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n − (cid:18) n + 2 ρ + ρ (cid:19) ˆ D ( ρ ) − ρ ˆ ∞ ρ t ˆ D ( t ) dt + O ( ρ − ) ˆ D ( ρ ) + O ( ρ − ) ˆ D ( ρ ) ˆ B ( ρ ) . (5.1) In particular, if B ( ρ ) > , ˆ N ′ ( ρ ) = − B ( ρ ) ˆ E ρ h∇ X ˆ h + N ( ρ )ˆ h, L − n ˆ h i Φ − n − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt − ρ ˆ B ( ρ ) ˆ S ρ |∇ X ˆ h + N ( ρ )ˆ h | |∇ r | Φ − n + O ( ρ − ) ˆ N ( ρ ) + O ( ρ − ) ˆ N ( ρ ) . (5.2) Proof.
The proof of Proposition 5.1 is a straightforward adaptation of the aforementionedresults due to Bernstein by using the corresponding Rellich-Necas identity for a C loc symmetric2-tensor ˆ h :div (cid:16) h∇ X ˆ h, ∇ · ˆ h i Φ − n (cid:17) := ∇ k (cid:16) ∇ X ˆ h ij ∇ k ˆ h ij Φ − n (cid:17) = ∇ k (cid:16) X l ∇ l ˆ h ij (cid:17) ∇ k ˆ h ij Φ − n + X l ∇ l ˆ h ij ∇ k (cid:16) ∇ k ˆ h ij Φ − n (cid:17) = ( ∇ k X l ) ∇ l ˆ h ij ∇ k ˆ h ij Φ − n + X l ∇ k ∇ l ˆ h ij ∇ k ˆ h ij Φ − n + X l ∇ l ˆ h ij ∇ k (cid:16) ∇ k ˆ h ij Φ − n (cid:17) = (cid:0) O ( r − ) (cid:1) |∇ ˆ h | Φ − n + h∇ X ∇ ˆ h, ∇ ˆ h i Φ − n + Rm( X , · , · , · ) ∗ ˆ h ∗ ∇ ˆ h Φ − n + h∇ X ˆ h, L − n ˆ h i Φ − n = |∇ ˆ h | Φ − n + h∇ X ∇ ˆ h, ∇ ˆ h i Φ − n + h∇ X ˆ h, L − n ˆ h i Φ − n + (cid:16) O ( r − ) |∇ ˆ h | + O ( r − ) | ˆ h ||∇ ˆ h | (cid:17) Φ − n . Here, we have used commutation formulas in the fourth line together with identity (2.17) toensure that Rm( g )( X , · , · , · ) = O ( r − ) in the fifth equality.By the very definition of ˆ D ( ρ ), one gets by integration by parts:ˆ D ′ ( ρ ) = − Φ − n ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | = 1 ρ ˆ E ρ div (cid:16) |∇ ˆ h | Φ − n X (cid:17) = 1 ρ ˆ E ρ (cid:18) ∇ X |∇ ˆ h | + (cid:18) − n − r (cid:19) |∇ ˆ h | (cid:19) Φ − n + O ( ρ − ) ˆ D ( ρ ) . Now, the co-area formula together with Fubini’s theorem, the divergence theorem andidentity (5) as in the proof of [Ber17, Proposition 4 .
2] give:ˆ D ′ ( ρ ) = 1 ρ ˆ E ρ − h∇ X ˆ h, L − n ˆ h i Φ − n − (cid:18) n + 2 ρ + ρ (cid:19) ˆ D ( ρ ) − ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n − ρ ˆ ∞ ρ t ˆ D ( t ) dt + O ( ρ − ) ˆ D ( ρ ) + O ( ρ − ) ˆ D ( ρ ) ˆ B ( ρ ) , where we have used the estimate from Proposition 4.1 to estimate the integral term involving | ˆ h | . This ends the proof of (5.1).The proof of (5.2) is word for word that of [Ber17, Corollary 4 .
3] based on (5.1). (cid:3)
We start with computing the derivative of the ”frequency” function associated to a vectorfield B satisfying (2.19), defined by N B ( ρ ) := ρ ´ S ρ |B| |∇ r | Ψ ˆ B ( ρ ) . Note that N B ( ρ ) = Ψ n +2 ( ρ )Ψ ( ρ ) B B ( ρ ) B ( ρ ) = Ψ ( ρ ) B B ( ρ )ˆ B ( ρ ) . Proposition 5.2.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Assume h is non-trivial.Then, for ρ ≥ R , N B ( ρ ) ≤ C (cid:0) ρ − N ( ρ ) + ρ − (cid:1) , (5.3) and N ′B ( ρ ) = (cid:18) ρ + 2 n + 4 ρ + O ( ρ − ) (cid:19) N B ( ρ ) + 2 N ( ρ ) ρ N B ( ρ )+ O ( ρ − ) ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) + O ( ρ − ) ! N B ( ρ ) . (5.4) Proof.
We begin with the proof of (5.3). From Lemma 4.2 we have N B ( ρ ) = ρ Ψ ( ρ ) B B ( ρ )ˆ B ( ρ ) ≤ C ˆ B ( ρ ) (cid:16) ρ − ˆ D ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) = C (cid:16) ρ − ˆ N ( ρ ) + ρ − (cid:17) ≤ C (cid:0) ρ − N ( ρ ) + ρ − (cid:1) , where we have used [(4.8), Lemma 4.3] in the last line.Now, by using Lemmata 2.8 and 3.5,ˆ B ( ρ ) N ′B ( ρ ) = ρ Ψ ( ρ ) B ′B ( ρ ) ˆ B ( ρ ) − ρ Ψ ( ρ ) B B ( ρ ) ˆ B ′ ( ρ )+ (cid:18) ρ ρ (cid:19) ρ Ψ ( ρ ) B B ( ρ ) ˆ B ( ρ )= (cid:18) ρ + 2 n + 4 ρ + O ( ρ − ) (cid:19) ρ Ψ ( ρ ) B B ( ρ ) ˆ B ( ρ ) + 2 ρ Ψ ( ρ ) ˆ F ( ρ ) B B ( ρ )+ ρ Ψ ( ρ ) ˆ S ρ O ( r − ) | h | + O ( r − ) |∇ h | ! B B ( ρ ) ˆ B ( ρ )= (cid:18) ρ + 2 n + 4 ρ + O ( ρ − ) (cid:19) ρ Ψ ( ρ ) B B ( ρ ) ˆ B ( ρ ) + 2 ρ Ψ ( ρ ) ˆ F ( ρ ) B B ( ρ )+ ˆ S ρ (cid:16) O ( r − ) | ˆ h | + O ( r − ) |∇ ˆ h | (cid:17) Φ − n ! (cid:0) ρ Ψ ( ρ ) B B ( ρ ) (cid:1) ˆ B ( ρ ) . RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 33
Dividing the previous estimate by ˆ B ( ρ ) (legitimated by Corollary 4.4) leads to the expectedresult. (cid:3) In order to estimate the derivative of the frequency function N , we need an additionallemma that handles the integral involving the Lie derivative of the Bianchi gauge on the levelsets of r : Lemma 5.3.
Let T be a C loc ( E R ) symmetric -tensor and let Y be a C loc ( E R ) vector field.Denote by g r the metric on S r induced by g and let T ⊤ (respectively Y ⊤ ) the tangential partof T (respectively Y ). Then, h T, L Y ( g ) i g = 2 [ N · h Y, T ( N ) i g − h Y, ∇ N N i g T ( N , N ) − N · ( T ( N , N )) h Y, N i g ]+ 2 h −h Y, ∇ N T ( N ) ⊤ i g − h Y ⊤ , ∇ T ( N ) ⊤ N i g + T ( N ) ⊤ · h Y, N i g i + h Y, N i g h T ⊤ , L N ( g ) i g + h T ⊤ , L Y ⊤ ( g r ) i g r . In particular, there is a positive constant C such that on E R , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r h T, L Y ( g ) i g − N · h Y, T ( N ) i g |∇ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S r | Y | g (cid:0) r − | T | g + |∇ T | g (cid:1) . Proof.
Decompose T as follows: T = T ( N , N ) d r ⊗ d r |∇ r | + h T ( N ) ⊤ , ·i g ⊗ d r |∇ r | + d r |∇ r | ⊗ h T ( N ) ⊤ , ·i g + T ⊤ . In particular, h T, L Y ( g ) i g = T ( N , N ) L Y ( g )( N , N ) + 2 L Y ( g )( N , T ( N ) ⊤ ) + h T ⊤ , L Y ( g ) i g = 2 T ( N , N ) h∇ N Y, N i g + 2 h∇ N Y, T ( N ) ⊤ i g + 2 h∇ T ( N ) ⊤ Y, N i g + h T ⊤ , L Y ( g ) i g = 2 [ N · ( T ( N , N ) h Y, N i g ) − h Y, ∇ N N i g T ( N , N ) − N · ( T ( N , N )) h Y, N i g ]+ 2 h N · h Y, T ( N ) ⊤ i g − h Y, ∇ N T ( N ) ⊤ i g + T ( N ) ⊤ · h Y, N i g − h Y ⊤ , ∇ T ( N ) ⊤ N i g i + h T ⊤ , L Y ( g ) i g = 2 [ N · h Y, T ( N ) i g − h Y, ∇ N N i g T ( N , N ) − N · ( T ( N , N )) h Y, N i g ]+ 2 h −h Y, ∇ N T ( N ) ⊤ i g + T ( N ) ⊤ · h Y, N i g − h Y ⊤ , ∇ T ( N ) ⊤ N i g i + h T ⊤ , L Y ( g ) i g . (5.5)Now, by decomposing Y = h Y, N i g N + Y ⊤ and g = |∇ r | d r ⊗ d r + g r , h T ⊤ , L Y ( g ) i g = h T ⊤ , L Y ⊤ ( g ) i g + h T ⊤ , L h Y, N i g N ( g ) i g = 1 |∇ r | h T ⊤ , L Y ⊤ ( d r ) ⊗ d r + d r ⊗ L Y ⊤ ( d r ) i g + h T ⊤ , L Y ⊤ ( g r ) i g r + h Y, N i g h T ⊤ , L N ( g ) i g + h T ⊤ , d ( h Y, N i g ) ⊗ h N , ·i g + h N , ·i g ⊗ d ( h Y, N i g ) i g = h T ⊤ , L Y ⊤ ( g r ) i g r + h Y, N i g h T ⊤ , L N ( g ) i g . (5.6) Estimates (5.5) and (5.6) lead to the expected identity.As for the integral estimate, notice that |∇ ( |∇ r | − ) | g = O ( r − ) and similarly, |L N ( g ) | g = O ( r − ) and |∇ N N | = O ( r − ). Use integration by parts with respect to the induced metric g r to obtain: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r h T, L Y ( g ) i g − N · h Y, T ( N ) i g |∇ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S r | Y | g (cid:0) r − | T | g + |∇ T | g (cid:1) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r h T ⊤ , L Y ⊤ ( g r ) i g r |∇ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r T ( N ) ⊤ · h Y, N i g |∇ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S r | Y | g (cid:0) r − | T | g + |∇ T | g (cid:1) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r h div g r (cid:16) |∇ r | − T ⊤ (cid:17) , Y ⊤ i g r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S r div g r (cid:16) |∇ r | − T ( N ) ⊤ (cid:17) · h Y, N i g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S r | Y | g (cid:0) r − | T | g + |∇ T | g (cid:1) + C ˆ S r (cid:18)(cid:12)(cid:12)(cid:12) div g r T ⊤ (cid:12)(cid:12)(cid:12) g r + (cid:12)(cid:12)(cid:12) div g r (cid:16) T ( N ) ⊤ (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) | Y | g . Now, since |∇ g r U V − ∇ gU V | ≤ C r − | U | g | V | g for any two vector fields U, V tangent to S r , onegets the expected integral estimate. (cid:3) We are in a position to state and prove the first main result of this section: the followingproposition establishes an a priori growth on the frequency function N ( ρ ). More precisely: Proposition 5.4. ([Ber17, Proposition 4.4])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Assume h is non-trivial.Then there exist positive constants C , ˜ C and ˜ R ≥ R such that if ρ ≥ ˜ R , (cid:16) N − ˜ CN B (cid:17) ′ ( ρ ) ≤ ρ − N ( ρ ) + Cρ − N ( ρ ) + C. (5.7) In particular, | N ( ρ ) | ≤ Cρ and N B ( ρ ) ≤ Cρ − for ρ ≥ R .Proof. Thanks to (2.25) together with Corollary 2.7, the co-area formula gives us:ˆ F ′ ( ρ ) = ˆ D ′ ( ρ ) − ˆ S ρ h ˆ h, L − n ˆ h i|∇ r | Φ − n = ˆ D ′ ( ρ ) − ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n − ˆ S ρ h ˆ h, R [ˆ h ] i|∇ r | Φ − n . (5.8)Observe that Cauchy-Schwarz inequality applied to the last integral on the righthand side of(5.8) gives: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S ρ h ˆ h, R [ˆ h ] i|∇ r | Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρ ˆ B ( ρ ) + Cρ ˆ B ( ρ ) ˆ S ρ |∇ ˆ h | Φ − n ! . (5.9) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 35
Now, according to [(5.1), Proposition 5.1] and Lemma 4.5,ˆ D ′ ( ρ ) ≤ − ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n − (cid:18) n + 2 ρ + ρ (cid:19) ˆ D ( ρ )+ Cρ (cid:16) ˆ D ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) + Cρ ˆ S ρ |∇ ˆ h | Φ − n ! ˆ S ρ r |B| Ψ ! ≤ − ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n − (cid:18) n + 2 ρ + ρ (cid:19) ˆ F ( ρ )+ 18 ρ (cid:16) ˆ F ( ρ ) + Cρ − ˆ B ( ρ ) (cid:17) + Cρ (cid:16) ˆ F ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) + Cρ ˆ S ρ |∇ ˆ h | Φ − n ! (cid:0) ρ Ψ ( ρ ) B B ( ρ ) (cid:1) . (5.10)Here we have used (2.25), together with [(4.7), Lemma 4.3] in the second inequality.Plugging (5.9) and (5.10) into (5.8) gives:ˆ F ′ ( ρ ) ≤ − ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n − (cid:18) n + 2 ρ + ρ (cid:19) ˆ F ( ρ )+ 18 ρ (cid:16) ˆ F ( ρ ) + Cρ − ˆ B ( ρ ) (cid:17) + Cρ (cid:16) ˆ F ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) + Cρ ˆ S ρ |∇ ˆ h | Φ − n ! "(cid:0) ρ Ψ ( ρ ) B B ( ρ ) (cid:1) + ˆ B ( ρ ) ρ − ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n . Now, thanks to Lemma 2.8,ˆ B ( ρ ) N ′ ( ρ ) = ρ ˆ F ′ ( ρ ) − ρ ˆ F ( ρ ) ˆ B ′ ( ρ )ˆ B ( ρ ) + ˆ F ( ρ ) ≤ − ρ ˆ S ρ |∇ N ˆ h | |∇ r | Φ − n + 2 ρ ˆ F ( ρ )ˆ B ( ρ ) + O ( ρ − ) ˆ F ( ρ )+ 18 (cid:16) ˆ F ( ρ ) + Cρ − ˆ B ( ρ ) (cid:17) + Cρ (cid:16) ˆ F ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) + C ˆ S ρ |∇ ˆ h | Φ − n ! "(cid:0) ρ Ψ ( ρ ) B B ( ρ ) (cid:1) + ˆ B ( ρ ) ρ − ρ ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n ≤ (cid:16) ˆ F ( ρ ) + Cρ − ˆ B ( ρ ) (cid:17) + Cρ (cid:16) ˆ F ( ρ ) + ρ − ˆ B ( ρ ) (cid:17) + C ˆ S ρ |∇ ˆ h | Φ − n ! "(cid:0) ρ Ψ ( ρ ) B B ( ρ ) (cid:1) + ˆ B ( ρ ) ρ − ρ ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n . In particular, dividing the previous estimate by ˆ B ( ρ ) gives us: N ′ ( ρ ) ≤ N ( ρ ) ρ + C N ( ρ ) ρ + C ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! (cid:18) N B ( ρ ) + 1 ρ (cid:19) + Cρ − ρ ˆ B ( ρ ) ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n . Invoking Lemma 5.3 applied to the symmetric 2-tensor ˆ h and the vector field B , one gets: N ′ ( ρ ) ≤ N ( ρ ) ρ + C N ( ρ ) ρ + C ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! (cid:18) N B ( ρ ) + 1 ρ (cid:19) + Cρ + C ρ − n ˆ B ( ρ ) ˆ S r |B| g (cid:16) r − | ˆ h | g + |∇ ˆ h | g (cid:17) − ρ − n ˆ B ( ρ ) ˆ S ρ N · hB , ˆ h ( N ) i g |∇ r | . (5.11)We note that using (2.19) we can write N · hB , ˆ h ( N ) i g = 2 r |∇ g r | h∇ ∇ f B , ˆ h ( N ) i g + hB , ∇ N ˆ h ( N ) i g + hB , ˆ h ( ∇ N N ) i g = O ( r − | ˆ h | + |∇ ˆ h | ) |B| + O ( r − | h | + r − |∇ h | ) | ˆ h | = O ( r − | ˆ h | + |∇ ˆ h | ) |B| + O ( r − | ˆ h | + r − |∇ ˆ h | ) | ˆ h | Φ − n . We can thus estimate ρ − n ˆ B ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S ρ N · hB , ˆ h ( N ) i g |∇ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρ − n − + Cρ − − n ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! + Cρ − n ˆ B ( ρ ) ˆ S r |B| g (cid:16) r − | ˆ h | + |∇ ˆ h | (cid:17) ≤ Cρ − + Cρ − ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! + C ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) + O ( ρ − ) ! N B ( ρ ) . (5.12)This implies, using (5.11), N ′ ( ρ ) ≤ N ( ρ ) ρ + C N ( ρ ) ρ + C ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! + O ( ρ − ) N B ( ρ )+ Cρ − + Cρ − ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! , where we have applied Cauchy-Schwarz inequality in the second line of (5.11). RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 37
Let us assume that ˆ B ( ρ ) ˆ S ρ |∇ ˆ h | Φ − n ≤ (cid:16) ˆ F ( ρ ) + ρ B ( ρ ) (cid:17) . (5.13)Using this assumption we arrive at N ′ ( ρ ) ≤ ρ − N ( ρ ) + Cρ − N ( ρ ) + Cρ − + C (cid:0) ρ − N ( ρ ) + ρ (cid:1) (cid:18) N B ( ρ ) + Cρ − (cid:19) ≤ ρ − N ( ρ ) + C + Cρ − N ( ρ ) N B ( ρ ) + CρN B ( ρ ) ≤ ρ − N ( ρ ) + C + Cρ − N ( ρ ) N B ( ρ ) + Cρ (cid:0) ρ − N ( ρ ) + ρ − (cid:1) ≤ ρ − N ( ρ ) + Cρ − N ( ρ ) + Cρ − N ( ρ ) N B ( ρ ) + C, (5.14)where C is a positive constant that may vary from line to line.Here we have used Young’s inequality in the second and last line, the bound (5.3) is usedin the third line and we have made constant use of the fact that N ( ρ ) ≥ − Cρ − thanks to(4.8). Now, under the assumption (5.13) and according to (5.4) and the bound (5.3), N ′B ( ρ ) ≥ − Cρ − N B ( ρ ) + 2 N ( ρ ) ρ N B ( ρ ) − C ρ − (cid:18) N ( ρ ) ρ + ρ (cid:19) + ρ − ! N B ( ρ ) ≥ − Cρ − N B ( ρ ) + 2 N ( ρ ) ρ N B ( ρ ) − C (cid:0) ρ − N ( ρ ) + ρ − (cid:1) N B ( ρ ) ≥ − Cρ − N B ( ρ ) + N ( ρ ) ρ N B ( ρ ) − Cρ − N ( ρ ) − Cρ − ≥ − Cρ − N ( ρ ) + N ( ρ ) ρ N B ( ρ ) − Cρ − . (5.15)Combining (5.14) together with (5.15) leads to:( N − CN B ) ′ ( ρ ) ≤ ρ − N ( ρ ) + Cρ − N ( ρ ) + C. (5.16)Assume now that (5.13) does not hold, i.e.ˆ B ( ρ ) ˆ S ρ |∇ ˆ h | Φ − n ≥ (cid:16) ˆ F ( ρ ) + ρ B ( ρ ) (cid:17) . (5.17)Going back to the identity (2.25) and using [(2.24), Corollary 2.7]:ˆ F ′ ( ρ ) = − ˆ S ρ |∇ ˆ h | |∇ r | Φ − n − ˆ S ρ h ˆ h, L B ( g ) i|∇ r | r − n − ˆ S ρ h ˆ h, R [ˆ h ] i|∇ r | Φ − n . Applying Lemma 5.3 to the symmetric 2-tensor ˆ h and the vector field B , and using (5.9),(5.12), ρ ˆ F ′ ( ρ )ˆ B ( ρ ) ≤ − ρ ˆ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − + Cρ − ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) ! + C ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) + O ( ρ − ) ! N B ( ρ ) ≤ − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − N B ( ρ ) + Cρ − N B ( ρ ) + Cρ − . Here we have used Young’s inequality in the second line. In particular, thanks to Lemma 2.8, N ′ ( ρ ) ≤ − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − (1 + ρ / N B ( ρ ) + N B ( ρ ))+ ρ ˆ F ( ρ )ˆ B ( ρ ) ρ n + 2 ρ + 2 ˆ F ( ρ )ˆ B ( ρ ) + O ( ρ − ) ! ≤ − ρ ˆ F ( ρ )ˆ B ( ρ ) + ρ ! − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − (1 + ρ / N B ( ρ ) + N B ( ρ ))+ ρ ˆ F ( ρ )ˆ B ( ρ ) ρ n + 2 ρ + 2 ˆ F ( ρ )ˆ B ( ρ ) + O ( ρ − ) ! = − ρ ˆ F ( ρ )ˆ B ( ρ ) ! − ˆ F ( ρ )ˆ B ( ρ ) (cid:0) ρ − ( n + 2) (cid:1) − ρ + O ( ρ − ) N ( ρ ) − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − (1 + ρ / N B ( ρ ) + N B ( ρ )) ≤ − ρ ˆ F ( ρ )ˆ B ( ρ ) ! − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n + Cρ − N ( ρ ) . (5.18)Here we have invoked assumption (5.17) in the second line and we have made constant use ofthe fact that N ( ρ ) ≥ − Cρ − thanks to (4.8). Finally the bound (5.3) is used in the last line.Therefore, for any positive constant ˜ C , one has thanks to (5.18) and (5.4): (cid:16) N − ˜ CN B (cid:17) ′ ( ρ ) ≤ − ρ ˆ F ( ρ )ˆ B ( ρ ) ! − ρ B ( ρ ) ˆ S ρ |∇ ˆ h | |∇ r | Φ − n − Cρ − N ( ρ ) N B ( ρ )+ Cρ − N ( ρ ) + ˜ C Cρ − ´ S ρ |∇ ˆ h | Φ − n ˆ B ( ρ ) + Cρ − ! N B ( ρ ) (5.19)+ Cρ − ≤ Cρ − N ( ρ ) + Cρ − , where C is a positive constant that may vary from line to line.Combining differential inequalities (5.16) and (5.19) give the first expected result (5.7).Integrating (5.7) between ρ and ρ gives in turn: N ( ρ ) ≤ ˜ CN B ( ρ ) + C (ˆ h, B , ρ ) + 14 ˆ ρρ N ( s ) s ds + C ˆ ρρ N ( s ) s ds + Cρ RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 39 ≤ ˆ ρρ N ( s ) s ds + C N ( ρ ) ρ + C (ˆ h, B , ρ ) + C ˆ ρρ N ( s ) s ds + Cρ, where we have used [(5.3), Proposition 5.2].In particular, if ρ is sufficiently large, N ( ρ ) ≤ C (ˆ h, B , ρ ) + 14 ˆ ρρ N ( s ) s ds + C ˆ ρρ N ( s ) s ds + Cρ.
Gr¨onwall’s inequality applied to the function ´ ρρ N ( s ) s ds give the expected a priori growth onthe frequency function N ( ρ ) ≤ Cρ for ρ ≥ R . Using [(5.3), Proposition 5.2] again proves thea priori bound on N B . (cid:3) Corollary 5.5. ([Ber17, Corollary 4.5])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) .Assume h is non-trivial.Then there exist positive constants C and ˜ R ≥ R such that if ρ ≥ ˜ R , ˆ N ′ ( ρ ) ≤ − ´ ∞ ρ t ˆ D ( t ) dt ˆ B ( ρ ) + Cρ − (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + 1 (cid:17) . (5.20) Moreover, ˆ N ′ ( ρ ) + 2 ρ ˆ N ( ρ ) ≤
11 + Cρ − ˆ N ( ρ ) − ρ ˆ N ( ρ ) − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt ! + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) . (5.21) Remark 5.6.
The reason of stating (5.21) is made clearer in the proof of Proposition 6.2 on thedecay of the frequency function ˆ N . Roughly speaking, one must keep track of the coefficient infront of the good term ´ ∞ ρ t ˆ D ( t ) dt : the coefficient in (5.20) is not enough for our purpose.Proof. According to [(5.2), Proposition 5.1],ˆ N ′ ( ρ ) ≤ − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt + 2ˆ B ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 | N ( ρ ) | ˆ B ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ E ρ h ˆ h, L − n ˆ h i Φ − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Cρ − ˆ N ( ρ ) + Cρ − ˆ N ( ρ ) . Thanks to [(4.6), Lemma 4.3] and Lemma 4.5,ˆ N ′ ( ρ ) ≤ − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) (cid:17) + C − ˆ D ′ ( ρ )ˆ B ( ρ ) ! N B ( ρ ) + Cρ − | N ( ρ ) | ˆ N ( ρ )+ Cρ − | N ( ρ ) | + Cρ − ˆ N ( ρ ) + Cρ − ˆ N ( ρ ) (5.22) ≤ − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) + C − ˆ D ′ ( ρ )ˆ B ( ρ ) ! N B ( ρ ) . Here we have used that due to (4.8) it holds that | N ( ρ ) | ≤ ˆ N ( ρ ) + O ( ρ − ). Note that by thevery definition of ˆ N : − ˆ D ′ ( ρ )ˆ B ( ρ ) ! = − (cid:16) ρ − ˆ B ˆ N (cid:17) ′ ( ρ )ˆ B ( ρ )= ρ − ˆ N ( ρ ) − ρ − ˆ N ′ ( ρ ) − ρ − ˆ B ′ ( ρ )ˆ B ( ρ ) ˆ N ( ρ ) ≤ − ρ − ˆ N ′ ( ρ ) + ρ − (cid:18) n + 2 ρ + ρ ρ N ( ρ ) + Cρ − (cid:19) ˆ N ( ρ ) ≤ − ρ − ˆ N ′ ( ρ ) + C ˆ N ( ρ ) . (5.23)Here we have used Lemma 2.8 in the third inequality together with Proposition 5.4.Now, thanks to [(5.3), Proposition 5.2], for ε > − ˆ D ′ ( ρ )ˆ B ( ρ ) ! N B ( ρ ) ≤ C − ˆ D ′ ( ρ )ˆ B ( ρ ) ! (cid:0) ρ − N ( ρ ) + ρ − (cid:1) ≤ ε − ˆ D ′ ( ρ )ˆ B ( ρ ) ! + Cε − ρ − ≤ Cε ˆ N ( ρ ) + Cε − ρ − − ερ − ˆ N ′ ( ρ ) , where we used (5.23) in the last line. Plugging this back into (5.22) we see that(1 + Cερ − ) ˆ N ′ ( ρ ) ≤ − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) + Cε ˆ N ( ρ ) + Cε − ρ − . (5.24)As a first result, let us choose ε := ρ − in (5.24) to get:ˆ N ′ ( ρ ) ≤ −
12 ˆ B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + 1 (cid:17) . This proves (5.20).
RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 41
Now, by introducing the term 2 ρ − ˆ N ( ρ ) artificially and by dividing the inequality (5.24)by 1 + Cερ − , one gets:ˆ N ′ ( ρ ) + 2 ρ + Cε ˆ N ( ρ ) ≤
11 +
Cερ − ρ ˆ N ( ρ ) − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt ! + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) + Cε ˆ N ( ρ ) + Cε − ρ − . Observe that:ˆ N ′ ( ρ ) + 2 ρ ˆ N ( ρ ) ≤
11 +
Cερ − ρ ˆ N ( ρ ) − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt ! + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) + 2 ρ (cid:18) −
11 +
Cερ − (cid:19) ˆ N ( ρ )+ Cε ˆ N ( ρ ) + Cε − ρ − ≤
11 +
Cερ − ρ ˆ N ( ρ ) − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt ! + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) + 2 Cερ ˆ N ( ρ )+ Cε ˆ N ( ρ ) + Cε − ρ − . Finally, let us choose ε := ρ − ˆ N − − n ( ρ ) (if ˆ N ( ρ ) = 0 then ˆ N ( ρ ′ ) = 0 for ρ ′ ≥ ρ and there isnothing to prove) to obtain:ˆ N ′ ( ρ ) + 2 ρ ˆ N ( ρ ) ≤
11 + Cρ − ˆ N − ( ρ ) ρ ˆ N ( ρ ) − B ( ρ ) ˆ ∞ ρ t ˆ D ( t ) dt ! + Cρ (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) + ρ − (cid:17) . This proves (5.21). (cid:3) Frequency decay
We start this section with the proof of the main preliminary result on the limiting behaviorof the frequency function ˆ N : Theorem 6.1. ([Ber17, Theorem 4.1])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) .Assume h is non-trivial. Then, lim ρ → + ∞ ˆ N ( ρ ) = lim ρ → + ∞ N ( ρ ) = 0 . Proof.
In case lim ρ → + ∞ ˆ N ( ρ ) = 0 then Lemma 4.3 implies lim ρ → + ∞ N ( ρ ) = 0.Now, [(5.20), Corollary 5.5] implies that the function ρ → e Cρ ( ˆ N ( ρ ) + 1) is non-increasingfor some large positive constant C . This already ensures the finiteness of the limit of ˆ N ( ρ ) as ρ tends to + ∞ , that we denote here by N ∞ . By the non-negativity of ˆ N , one has N ∞ ≥ Then to show that N ∞ = 0 goes along the same lines as in the proof of [Ber17, Theorem 4.1]using Lemma 2.9. (cid:3) The next proposition discretizes the problem of showing some decay on the frequencyfunction.
Proposition 6.2. ([Ber17, Proposition 5.4])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Assume h is non-trivial and assume ˆ N ( ρ ) ≤ ηρ γ , (6.1) for ρ ≥ R and some γ ∈ [ − , . Then there exist constants R ′ ≥ R and C ≥ , depending on h , so that, for ρ ≥ R ′ , either (1) ˆ N ( ρ + 2) − ˆ N ( ρ ) ≤ − ρ − ˆ N ( ρ ) , or (2) ˆ N ( ρ + 1) − ˆ N ( ρ ) ≤ − ρ − ˆ N ( ρ ) + Cρ − γ . (6.2) Proof.
By (5.20) and Theorem 6.1 and γ ∈ [ − , R ′ ≥ R and κ ≥ s ≥ R ′ together with (6.1) implies thatˆ N ′ ( s ) ≤ Cs (cid:16) ˆ N ( s ) + ˆ N ( s ) + s − (cid:17) ≤ κs − γ . As γ ≤
0, the mean value inequality gives for any τ ∈ [0 ,
2] and s ≥ R ′ ,ˆ N ( s + τ ) ≤ ˆ N ( s ) + 15 κs − γ . (6.3)Suppose case (1) does not hold for a given ρ , that isˆ N ( ρ + 2) − ˆ N ( ρ ) > − ρ − ˆ N ( ρ ) . We first claim that for all τ ∈ [0 , N ( ρ + τ ) − ˆ N ( ρ ) ≥ − ρ − ˆ N ( ρ ) − κρ − γ . (6.4)Indeed if (6.4) fails for some τ ∈ [0 , s = ρ + τ ˆ N ( ρ + 2) = ˆ N ( ρ + τ + (2 − τ )) ≤ ˆ N ( ρ + τ ) + 15 κ ( ρ + τ ) − γ ≤ ˆ N ( ρ + τ ) + 15 κρ − γ < ˆ N ( ρ ) − ρ − ˆ N ( ρ ) + 15 κρ − γ − κρ − γ < ˆ N ( ρ ) − ρ − ˆ N ( ρ ) . This contradicts the assumption that case (1) does not hold at ρ , proving the claim.If R ′ is sufficiently large, Lemma 2.9 together with the fact that ˆ N ( ρ ) ≤ η/ ρ ≥ R ′ guarantees that for t ∈ [ ρ, ρ + 2], (1 − ηρ − ) B ( ρ ) ≤ B ( t ) . Thus by (6.4), if t ∈ [ ρ, ρ + 2], tD ( t ) ≥ (cid:18) − ηρ (cid:19) ˆ N ( t ) B ( ρ ) ≥ (cid:18) − ηρ (cid:19) (cid:18)(cid:18) − ρ (cid:19) ˆ N ( ρ ) − κρ − γ (cid:19) B ( ρ ) ≥ (1 − (2 η + 4) ρ − )) ˆ N ( ρ ) B ( ρ ) − κρ − γ B ( ρ ) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 43
Hence, using (4.2) and (6.1) we see that for C sufficiently large, ˆ ∞ ρ t ˆ D ( t ) dt ≥ ˆ ρ +1 ρ tD ( t )Φ − n ( t ) dt ≥ (cid:18) − η + 4 ρ (cid:19) ˆ N ( ρ ) B ( ρ ) ˆ ρ +1 ρ Φ − n ( t ) dt − κρ − γ B ( ρ ) ˆ ρ +1 ρ Φ − n ( t ) dt ≥ ρ ˆ N ( ρ ) ˆ B ( ρ ) − Cρ − γ ˆ B ( ρ ) . The previous estimate together with [(5.21), Corollary 5.5] and (6.1) yieldsˆ N ′ ( ρ ) + 2 ρ ˆ N ( ρ ) ≤ Cρ − γ , for some positive constant C that may vary from line to line.Note that in the last estimate ρ can be replaced by ρ + τ for any τ ∈ [0 , τ ∈ [0 , N ′ ( ρ + τ ) ≤ − ρ + τ ) − ˆ N ( ρ + τ ) + C ( ρ + τ ) − γ ≤ − ρ − ˆ N ( ρ + τ ) + Cρ − γ ≤ − ρ − ˆ N ( ρ ) + 8 ρ − ˆ N ( ρ ) + Cρ − γ ≤ − ρ − ˆ N ( ρ ) + Cρ − γ , which leads to the second estimate [(2), (6.2)] after integration between 0 and 1. (cid:3) We are now in a position to prove the main result of this section:
Theorem 6.3. ([Ber17, Theorem 5.1])
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) .Assume h is non-trivial. Then for ε > , there exists a positive constant C ε such that if ρ ≥ R : ˆ N ( ρ ) ≤ C ε ρ − ε . Proof.
We proceed as in the proof of [Ber17, Theorem 5 . N ( ρ ) := ρ ˆ N ( ρ ).We assume that (6.1) holds. Assume Case (1) of Proposition 6.2 holds true at ρ then astraightforward computation shows: N ( ρ + 2) − N ( ρ ) ≤ . Now, if Case (2) of Proposition 6.2 holds then: N ( ρ + 1) − N ( ρ ) = ( ρ + 1) ˆ N ( ρ + 1) − ρ ˆ N ( ρ ) ≤ − ρ ˆ N ( ρ ) + Cρ γ + 2 ρ ˆ N ( ρ + 1) + ˆ N ( ρ + 1)= 2 ρ (cid:16) ˆ N ( ρ + 1) − ˆ N ( ρ ) (cid:17) + ˆ N ( ρ + 1) + Cρ γ ≤ ρ (cid:16) − ρ − ˆ N ( ρ ) + Cρ − γ (cid:17) + ˆ N ( ρ + 1) + Cρ γ ≤ Cρ γ , where C is a positive constant that may vary from line to line.For each i ≥
0, let min ρ ∈ [ R + i,R + i +2] N ( ρ ) =: N − i ≤ N + i := max ρ ∈ [ R + i,R + i +2] N ( ρ ) . According to Theorem 6.1, there is an η > γ = 0. Therefore, ineither case of Proposition 6.2, one gets: N + i +1 ≤ N + i + C. In particular, if ρ ∈ [ R + i, R + i + 2], then ρ ˆ N ( ρ ) ≤ N + i ≤ N +0 + Cρ.
This shows (6.1) holds true with γ = − .Injecting this improved decay back to Case (2) of Proposition 6.2, the previous reasoninggives us: N + i +1 ≤ N + i + C √ R + i , i ≥ . This leads us to: ρ ˆ N ( ρ ) ≤ N +0 + Cρ , for ρ ≥ R , i.e. (6.1) holds true with γ = − .Given ε ∈ (0 , N by iterating finitely many times theprevious arguments. (cid:3) Decay estimates and traces at infinity
Existence of traces at infinity.
The main result of this section is the following theorem which corresponds to ([Ber17,Theorem 6.1]).
Theorem 7.1.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then the following asser-tions hold true.(1) The limit lim → + ∞ ρ − n B ( ρ ) exists and is zero if and only if h ≡ on E R .(2) There is a positive constant C such that for ρ ≥ R , ˆ E ρ | ˆ h | r − − n ≤ Cρ n ˆ S ρ | ˆ h | , and for all ε ∈ (0 , , there is a positive constant C ε and a radius R ε ≥ R such thatthe following decay estimates hold for ρ ≥ R ε : ˆ E ρ (cid:16) r |∇ ˆ h | + r |∇ ∂ r ˆ h | (cid:17) r − − n ≤ C ε ρ n − ε ˆ S ρ | ˆ h | . (7.1) (3) Moreover, the radial limit lim r → + ∞ ˆ h =: tr ∞ ˆ h exists in the L ( g S ) -topology and: ˆ E ρ (cid:16) r (cid:16) | ˆ h − tr ∞ ˆ h | + |∇ ˆ h | (cid:17) + r |∇ ∂ r ˆ h | (cid:17) r − − n ≤ C ε k tr ∞ ˆ h k L ( S ) ρ − ε , ρ ≥ R ε . (7.2) Remark 7.2.
Notice that the integral estimate on ˆ h is optimal and the integral gradient estimateis sharp up to ε . This explains that we cannot guarantee the trace at infinity tr ∞ ˆ h to be anelement of H ( S ) unlike in [Ber17] . RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 45
Remark 7.3.
The integral radial estimate does not seem to be sharp since ˆ h is asymptotically -homogeneous, i.e. the k -th covariant derivatives of ˆ h are expected to decay like r − k at leastin the integral sense for k ≥ if the asymptotic cone is smooth outside the origin. Then by( (2.18) , (2.19) ), one can see that the radial derivative of ˆ h should decay like r − . Nonetheless,estimate (7.1) will be sufficient for our applications. Remark 7.4. If ( C ( S ) , dr + r g S , o ) is Ricci flat then it can be shown that the trace at infinity of ˆ h defines a smooth symmetric -tensor on the link S which is radially invariant: see [Der17b] . Remark 7.5.
As a final remark, thanks to Section 2.3, Theorem 7.1 can be applied to eachmetric g σ and potential function f σ , σ ∈ [1 , , and the estimates are uniform in σ ∈ [1 , .This fact will be needed in Section 8.Proof. According to Lemma 2.8 and Theorem 6.3, for any ε ∈ (0 , ddρ ρ − n B ( ρ ) = O ε ( ρ − ε ) ρ − n B ( ρ ) , where O ε depends on ε . Since the function ρ → ρ − ε is integrable on the half-line [ R, + ∞ ),equation (7.1) tells us that the limit lim ρ → + ∞ ρ − n B ( ρ ) =: B ∞ exists. Moreover, B ∞ = 0 ifand only if B ( ρ ) = 0 for some ρ ≥ R if and only if h ≡ E R thanks to Corollary 4.4.From now on, we assume B ∞ >
0, i.e. h is non-trivial on E R . By the co-area formula, if R is large enough so that 2 − B ∞ ≤ ρ − n +1 B ( ρ ) ≤ B ∞ for ρ ≥ R : ˆ E ρ | ˆ h | r − − n ≤ ˆ + ∞ ρ r − − n B ( r ) dr ≤ B ∞ ˆ + ∞ ρ r − dr ≤ ρ − n B ( ρ ) . Let us estimate the covariant derivatives of ˆ h as follows: by Theorem 6.3, D ( ρ ) = ρ − ˆ N ( ρ ) B ( ρ ) ≤ C ε B ∞ ρ n − ε , ρ ≥ R. (7.3)Differentiating the function ρ − n D ( ρ ) by using the co-area formula only: ddρ (cid:0) ρ − n D ( ρ ) (cid:1) = (cid:18) ρ n + 1 ρ (cid:19) ρ − n D ( ρ ) − ρ − n ˆ S ρ |∇ ˆ h | |∇ r | . (7.4)In particular, by integrating the previous relation between ρ ≥ R and + ∞ and by invoking(7.3), ˆ E ρ r − n |∇ ˆ h | ≤ C ε B ∞ ρ − ε ≤ C ε ρ − n + ε B ( ρ ) , ρ ≥ R, where C ε is a positive constant that may vary from line to line.Let us now prove the integral estimate on the radial derivative of ˆ h in (7.1).According to Proposition 5.1,( ρ − n D ) ′ ( ρ ) = − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | − ρ − n Φ − − n ( ρ ) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n + ρ − n D ( ρ ) − ρ − n Φ − − n ( ρ ) ˆ + ∞ ρ t ˆ D ( t ) dt + O ( ρ − ) ρ − n D ( ρ ) + O (cid:16) ρ − − n Φ − − n ( ρ ) (cid:17) ˆ D ( ρ ) ˆ B ( ρ ) ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | − ρ − n Φ − − n ( ρ ) ˆ E ρ h∇ X ˆ h, L − n ˆ h i Φ − n + C ε B ∞ ρ − ε + O (cid:16) ρ − − n Φ − − n ( ρ ) (cid:17) ˆ D ( ρ ) ˆ B ( ρ ) . Here we have used (7.3) in the last line.Now, thanks to Lemma 4.5,( ρ − n D ) ′ ( ρ ) ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | + Cρ − n Φ − − n ( ρ ) ˆ D ( ρ ) + 1 ρ ˆ D − n ( ρ ) ˆ B − n ( ρ ) ! + Cρ − n Φ − − n ( ρ ) ˆ S ρ |∇ ˆ h | Φ − n ! ˆ S ρ r |B| Ψ ! + C ε B ∞ ρ − ε + Cρ − − n Φ − − n ( ρ ) ˆ D ( ρ ) ˆ B ( ρ ) ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | + Cρ − n (cid:16) ˆ N ( ρ ) + ˆ N ( ρ ) (cid:17) B ( ρ )+ Cρ − n N B ( ρ ) B ( ρ ) ˆ S ρ |∇ ˆ h | ! + C ε B ∞ ρ − ε ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | + C ε B ∞ ρ − ε + Cρ − n ˆ S ρ |∇ ˆ h | ! B ( ρ ) , (7.5)where we have used N B ( ρ ) = O ( ρ − ) in the last line thanks to Proposition 5.4 together withTheorem 6.3 and the upper bound on B ( ρ ). In order to handle the energy of ˆ h on the levelset S ρ , we use (7.4) once more to deduce that:12 ˆ S ρ |∇ ˆ h | ≤ ρD ( ρ ) − ρ n − ( ρ − n D ) ′ ( ρ ) ≤ C ε B ∞ ρ n − ε − ρ n − ( ρ − n D ) ′ ( ρ ) , where we have used (7.3) in the second line. In particular, by injecting this estimate back to(7.5) and using Young’s inequality give us for any η > ρ − n D ) ′ ( ρ ) ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | + C ε B ∞ ρ − ε + η (cid:0) C ε B ∞ ρ − ε − Cρ − ( ρ − n D ) ′ ( ρ ) (cid:1) + CB ∞ η − ρ − . Therefore, by choosing η := ρ − ε , we get for ρ sufficiently large so that 1 + Cρ − − ε ≤ ρ − n D ) ′ ( ρ ) ≤ − ρ − n ˆ S ρ |∇ N ˆ h | |∇ r | + C ε B ∞ ρ − ε . (7.6)Integrating (7.6) between ρ and + ∞ gives the expected integral decay on the radial derivativeof ˆ h by noticing that lim ρ → + ∞ ρ − n D ( ρ ) = 0 by (7.3) for ε ∈ (0 , h exists in the L ( S )-topology followdirectly from the arguments in the proof of [Ber17, Appendix A ], see also [Ber17, Theorem6 . (cid:3) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 47
Proposition 7.6.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then for any ε ∈ (0 , ,there is a positive constant C ε and a radius R ε ≥ R such that the following decay estimatehold for ρ ≥ R ε : ˆ E ρ r (cid:12)(cid:12)(cid:12)(cid:12) ˆ h ( ∂ r ) − tr ˆ h ∂ r (cid:12)(cid:12)(cid:12)(cid:12) r − − n ≤ C ε ρ n − ε ˆ S ρ | ˆ h | . (7.7) In particular, for any ε ∈ (0 , , there exist a positive constant C ε and a diverging sequence ( ρ i ) i such that: ρ − n +1 i ˆ S ρi (cid:12)(cid:12)(cid:12)(cid:12) ˆ h ( ∂ r ) − tr ˆ h ∂ r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε ρ − (4 − ε ) i , (7.8) and, (cid:0) tr ∞ ˆ h (cid:1) ( ∂ r ) = tr g C (tr ∞ ˆ h )2 ∂ r , on the link S .Proof. Observe that B = div h − ∇ tr h − n (cid:18) div ˆ h − ∇ tr ˆ h (cid:19) − Ψ − n (cid:18) n r (cid:19) (cid:18) ˆ h ( r ∂ r ) − tr ˆ h r ∂ r (cid:19) . In particular, if ρ ≥ R , ˆ S ρ (cid:12)(cid:12)(cid:12)(cid:12) ˆ h ( ∂ r ) − tr ˆ h ∂ r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S ρ r − |B| Ψ n + Cρ − ˆ S ρ |∇ ˆ h | ≤ Cρ − N B ( ρ ) B ( ρ ) + Cρ − ˆ S ρ |∇ ˆ h | ≤ Cρ − B ( ρ ) + Cρ − ˆ S ρ |∇ ˆ h | , where C is a positive constant that may vary from line to line. Here we have used Proposition5.4 in the last line. Since B ( ρ ) ≤ B ∞ ρ n − for ρ ≥ R , one gets immediately: ρ − n +1 ˆ S ρ (cid:12)(cid:12)(cid:12)(cid:12) ˆ h ( ∂ r ) − tr ˆ h ∂ r (cid:12)(cid:12)(cid:12)(cid:12) ≤ CB ∞ ρ − + Cρ − − ( n − ˆ S ρ |∇ ˆ h | , ρ ≥ R. Multiplying this inequality by ρ and integrating from ρ to + ∞ leads to the proof of (7.7)once [(7.1), Theorem 7.1] is invoked. Given ε ∈ (0 , ρ i ) i such that: ˆ S ρi |∇ ˆ h | ≤ C ε ρ n − εi . This proves (7.8). In particular, this ensures thatlim ρ i → + ∞ ρ − n +1 i ˆ S ρi (cid:12)(cid:12)(cid:12)(cid:12) ˆ h ( ∂ r ) − tr ˆ h ∂ r (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Since the convergence of ˆ h to its trace at infinity tr ∞ ˆ h holds in the L ( S )-topology, theexpected result follows. (cid:3) Proposition 7.7.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then, div g S (tr ∞ ˆ h ) ⊤ = 0 , in the weak sense. Proof.
Define the difference of the Ricci tensors by R := Ric( g ) − Ric( g ) and define therescaled difference of the Ricci curvatures by ˆ R := Ψ n − R . We will use the notation g for g as the reference metric in the sequel. Claim 7.8.
The radial limit lim r → + ∞ ˆ R =: tr ∞ ˆ R exists in the L ( S ) -topology and, tr ∞ ˆ R = −
18 tr ∞ ˆ h. Moreover, on E R , there is some positive constant C such that: | R + ˆ h | ≤ C (cid:16) r − | ˆ h | + r − |∇ ∂ r ˆ h | (cid:17) . (7.9) Proof of Claim 7.8.
By using the soliton equation (2.2) satisfied by both metrics g = g and g : 2 ˆ R = 2Ψ n − (Ric( g ) − Ric( g )) = Ψ n − ( − h + L ∇ f h )= − r − ˆ h + Ψ n − L ∇ f (cid:16) Φ − n ˆ h (cid:17) = r − (cid:16) − ˆ h + L ∇ f ˆ h (cid:17) − (cid:18)
14 + n r (cid:19) |∇ r | ˆ h = −
14 ˆ h + O ( r − )ˆ h + O ( r − ) ∇ ∂ r ˆ h, (7.10)where we have used (2.7) and (2.8) together with the fact that L ∇ f ˆ h = ∇ ∇ f ˆ h + 12 (cid:16) ˆ h ◦ L ∇ f ( g ) + L ∇ f ( g ) ◦ ˆ h (cid:17) . By Theorem 7.1 and (7.10), we obtain the expected claim. (cid:3)
In particular, Claim 7.8 ensures that it suffices to prove that div g S (tr ∞ ˆ R ) ⊤ = 0 in theweak sense.To do so, we work on the end E R first and then we pass to the limit. Claim 7.9. (cid:12)(cid:12)(cid:12)(cid:12) div ˆ R + ( n − r ˆ h ( ∂ r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) r − | ˆ h | + r − |∇ ∂ r ˆ h | + r − |∇ ˆ h | (cid:17) Proof of Claim 7.9.
By invoking (2.4) in the following form div g Ric( g ) + Ric( g )( ∇ r /
4) = 0valid for the metric g as well,div g ˆ R = div g (Ψ n − (Ric( g ) − Ric( g )))= Ψ n − div g R + (cid:18) n − r (cid:19) ˆ R (cid:18) ∇ (cid:18) r (cid:19)(cid:19) = Ψ n − (div g − div g ) Ric( g ) + 2( n − r ˆ R (cid:18) ∇ (cid:18) r (cid:19)(cid:19) = ( n − r ˆ R ( ∂ r ) + Ψ n − (cid:0) O ( r − ) | h | + O ( r − ) |∇ h | (cid:1) = ( n − r ˆ R ( ∂ r ) + O ( r − ) | ˆ h | + O ( r − ) |∇ ˆ h | . In particular, [(7.9), Claim 7.8] gives the expected result. (cid:3)
RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 49
Let V be a vector field on the link S of the cone that we extend radially to the end E R sothat [ ∇ f, V ] = 0. Let g r be the metric on S r induced by g . Notice that the norm of V withrespect to g grows linearly as r tends to + ∞ and that h V, N i g = ( g − g C )( V, N ) = O ( r − ) | V | g = O ( r − ) . (7.11) Claim 7.10. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ S ρ (cid:10) div g ρ ˆ R ⊤ , V ⊤ (cid:11) g ρ + N · ˆ R ( N , V ⊤ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S ρ (cid:16) | ˆ h ⊤ ( ∂ r ) | + r − | ˆ h | + r − |∇ ∂ r ˆ h | + r − |∇ h | (cid:17) . Proof of Claim 7.10.
Let ( e i ) i =1 ,..., n − be an orthonormal frame for the metric g ρ . By defini-tion of the divergence of a symmetric 2-tensor, (cid:10) div g ˆ R , V ⊤ (cid:11) = (cid:10) ∇ N ˆ R N , V ⊤ (cid:11) + X e i ⊥ N (cid:10) ∇ e i ˆ R e i , V ⊤ (cid:11) = (cid:10) ∇ N ˆ R N , V ⊤ (cid:11) + X e i ⊥ N (cid:10) ∇ e i ˆ R ⊤ e i , V ⊤ (cid:11) + (cid:10) ∇ e i (cid:0) ˆ R ( N ) ⊗ N + N ⊗ ˆ R ( N ) (cid:1) e i , V ⊤ (cid:11) . (7.12)Now, since h V ⊤ , N i g = 0, (cid:12)(cid:12)(cid:12)(cid:12) X e i ⊥ N D ∇ e i (cid:16) ˆ R ( N ) ⊗ N + N ⊗ ˆ R ( N ) (cid:17) e i , V ⊤ E(cid:12)(cid:12)(cid:12)(cid:12) ≤ C r − | ˆ R ( N ) ⊤ || V | ≤ C | ˆ R ( N ) ⊤ | , (7.13)and by using the fact that ∇ ge i e j = ∇ g r e i e j − r − (cid:0) δ ij + O ( r − ) (cid:1) N together with the definitionof the divergence of a symmetric 2-tensor: X e i ⊥ N (cid:10) ∇ e i ˆ R ⊤ e i , V ⊤ (cid:11) = X e i ⊥ N e i · ˆ R ⊤ ( e i , V ⊤ ) − ˆ R ⊤ ( ∇ ge i e i , V ⊤ ) − ˆ R ⊤ ( e i , ∇ ge i V ⊤ )= X e i ⊥ N e i · ˆ R ⊤ ( e i , V ⊤ ) − ˆ R ⊤ ( ∇ g r e i e i , V ⊤ ) − ˆ R ⊤ ( e i , ∇ g r e i V ⊤ )+ O ( r − ) | ˆ R ( N ) ⊤ || V ⊤ | = (cid:10) div g r ˆ R ⊤ , V ⊤ (cid:11) g r + O ( r − ) | ˆ R ( N ) ⊤ || V ⊤ | , (7.14)Finally, since [ V, ∇ f ] = 0, (2.7) and (7.11) give, ∇ N V ⊤ = ∇ N ( V − h V, N i N )= 1 |∇ f | ∇ V ∇ f − N · h V, N i N − h V, N i∇ N N = r − V + O ( r − ) = r − V ⊤ + O ( r − ) , and, (cid:10) ∇ N ˆ R N , V ⊤ (cid:11) = N · ˆ R ( N , V ⊤ ) − ˆ R ( ∇ N N , V ⊤ ) − ˆ R ( N , ∇ N V ⊤ )= N · ˆ R ( N , V ⊤ ) + O ( r − ) | ˆ R| + O ( r − ) | ˆ R ( N ) ⊤ || V ⊤ | , (7.15)where we have used (2.7) in the second line.Injecting (7.13), (7.14) and (7.15) back to (7.12) gives the expected claim, once we invoke[(7.9), Claim 7.8] and Claim 7.9. (cid:3) We are in a position to conclude the proof of Proposition 7.7.On the one hand, by integration by parts on the level sets together with Claim 7.8: (cid:12)(cid:12)(cid:12)(cid:12) ˆ S ρ (cid:10) div g ρ ˆ R ⊤ , V ⊤ (cid:11) g ρ − (cid:10) ˆ h ⊤ , L V ⊤ ( g ρ ) (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ S ρ r − | ˆ h | + r − |∇ ∂ r ˆ h | . (7.16)On the other hand, by the co-area formula and integration by parts: ˆ ρ ρ ρ − n +1 ˆ S ρ N · ˆ R ( N , V ⊤ ) dρ = ˆ A ρ ,ρ (cid:10) ∇ r , ∇ ˆ R ( N , V ⊤ ) (cid:11) r − n +1 = ˆ S ρ |∇ r | ˆ R ( N , V ⊤ ) r − n +1 (cid:12)(cid:12)(cid:12)(cid:12) ρ ρ − ˆ A ρ ,ρ div( r − n +1 ∇ r ) ˆ R ( N , V ⊤ ) . (7.17)In particular, by combining Claims 7.8 and 7.10 together with (7.16) and (7.17): ˆ ρ ρ ρ − n +1 (cid:12)(cid:12)(cid:12)(cid:12) ˆ S ρ h ˆ h ⊤ , L V ⊤ ( g ρ ) i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ A ρ ,ρ (cid:16) | ˆ h ⊤ ( ∂ r ) | + r − | ˆ h | + r − |∇ ∂ r ˆ h | + r − |∇ h | (cid:17) r − n +1 + Cρ − n +21 ˆ S ρ | ˆ h ⊤ ( ∂ r ) | + r − | ˆ h | + r − |∇ ∂ r ˆ h | + Cρ − n +22 ˆ S ρ | ˆ h ⊤ ( ∂ r ) | + r − | ˆ h | + r − |∇ ∂ r ˆ h | . (7.18)Now, on the one hand, according to [(7.1), Theorem 7.1] and [(7.7), (7.8), Proposition 7.6],the righthand side of (7.18) is bounded as ρ tends to + ∞ , at least sequentially. On the otherhand, the integrand on the lefthand side converges to a finite value: more precisely,lim ρ → + ∞ ρ − n +1 ˆ S ρ (cid:10) ˆ h ⊤ , L V ⊤ ( g ρ ) (cid:11) = ˆ S (cid:10) tr ∞ ˆ h ⊤ , L V ( g S ) (cid:11) , where ( S, g S ) is the link of the asymptotic cone of both expanders. These two remarks togetherwith (7.18) imposes the vanishing of the previous limit. This ends the proof of Proposition7.7. (cid:3) As a corollary of Theorem 7.1, we get the following non-optimal pointwise decay in spacefor higher derivatives of the difference of two expanding gradient Ricci solitons coming out ofthe same cone:
Corollary 7.11.
Let ( h, B ) ∈ C ( E R ) satisfy (2.18) – (2.20) and (4.1) . Then for any η > and k ≥ , there exists a positive constant C η,k such: |∇ k h | ≤ C η,k e − (1 − η ) r , on E R . Proof.
The proof is a straightforward application of the following Gagliardo-Nirenberg inter-polation inequalities.Let j ∈ N and k ∈ N such that k > j and define α ∈ (0 ,
1) by α := j + n k + n . Then for anytensor T compactly supported with support in a ball B ( x, i ) for some point x ∈ M with i strictly less than half the injectivity radius of the metric g , k∇ j T k C ≤ C k∇ k T k αC k T k − αL , (7.19)for some positive constant C independent of the point x ∈ M .Let x ∈ E R such that B ( x, i ) is compactly contained in E R and let ϕ : M → [0 ,
1] be asmooth cut-off function such that its support is in B ( x, i ) and such that ϕ ≡ B ( x, i ).Then let us apply (7.19) to T := ϕ · h to get: k∇ j h k C ( B ( x,i )) ≤ C k h k αC k ( B ( x, i )) k h k − αL ( B ( x, i )) ≤ C j,k ( h ) k h k − αL ( B ( x, i )) . Here we have used the fact that h and its covariant derivatives are bounded on E R by Theorem2.1.Now, the fact that B ( x, i ) ⊂ E r ( x ) − C for some uniform positive constant C = C ( i )together with Theorem 7.1 give us for any positive ε > k∇ j h k C ( B ( x,i )) ≤ C j,k ( h ) (cid:18) ˆ E r ( x ) − C | h | (cid:19) − α = C j,k ( h ) (cid:18) ˆ E r ( x ) − C | ˆ h | r − n e − r (cid:19) − α ≤ C j,k ( h ) e − (1 − α ) ( r ( x ) − C )24 ≤ C j,k,ε ( h ) e − (1 − α − ε ) r ( x )24 (7.20)Given η >
0, then (7.20) shows the expected estimate once k (and hence α ) and ε are fixedso that 1 − α − ε ≥ − η . (cid:3) Asymptotically conical K¨ahler expanding gradient Ricci solitons.
We illustrate Theorem 7.1 in the context of K¨ahler expanding gradient Ricci solitons.Recall that if ( M n , J, ω ) is K¨ahler with K¨ahler form ω and complex structure J and if g denotes the Riemannian metric associated to ω , then we say that ( M n , J, ω, ∇ g f ) is agradient K¨ahler expanding Ricci soliton if ∇ g f is real holomorphic and ρ ω − i∂∂f = − ω , where ρ ω is the Ricci form of ω , i.e. ρ ω ( u, v ) := Ric( g )( J u, v ) for u, v ∈ T M .Now, recall that a K¨ahler cone is a Riemannian cone C (we omit the reference to the linkhere) such that the cone metric g C is K¨ahler together with a choice of g C -parallel complexstructure J C . We then have a K¨ahler form ω C ( u, v ) := g C ( J u, v ) for u, v ∈ T ( C \ { } ) and ω C = i ∂∂r with respect to J C . The vector field r∂ r is real holomorphic and the vector field J C r∂ r , called the Reeb vector field, is real holomorphic and Killing.Given a K¨ahler cone ( C, J C , ω C = i ∂∂r ), the level set { r = 1 } is traditionally denoted by S (standing for Sasakian) endowed with its induced Riemannian metric g S . The restrictionof the Reeb vector field to S induces a non-zero vector field ξ := J C r∂ r | r =1 on S . If η denotesthe g S -dual one-form of ξ then we get the g S orthogonal decomposition T S = D ⊕ R ξ , where D is the kernel of η . At the level of metrics, one gets g S = η ⊗ η + g T where g T := g S | D . Wecall g T the transverse metric. With these definitions in hand, Corollary 7.12.
Let ( M n , J, ω, ∇ g f ) and ( M n , J, ω ′ , ∇ g ′ f ′ ) be two K¨ahler expanding gradientRicci solitons asymptotic to the same K¨ahler cone ( C, J C , ω C , r∂ r / . Then the trace at infin-ity of the difference of the Ricci tensors ˆ R preserves both the radial vector field ∂ r and J ∂ r .More precisely, tr ∞ ˆ R = tr g C (tr ∞ R )2 (cid:0) dr + r η ⊗ η (cid:1) + Ric T ∞ , where Ric T ∞ is a symmetric tensor that preserves D . In particular, Ric T ∞ is trace free anddivergence free, i.e. tr g C Ric T ∞ ( g ) = 0 and div g T Ric T ∞ = 0 in the weak sense. Notice that under the assumptions of Corollary 7.12, it has been proved in [CDS19] that( M n , J, ω, ∇ g f ) is the unique (up to pullback by biholomorphisms) complete expanding gra-dient K¨ahler-Ricci soliton asymptotic to the K¨ahler cone ( C, J C , ω C , r∂ r / M isthe smooth canonical model of C , and there exists a resolution map π : M → C such that dπ ( ∇ g f ) = r∂ r : see Theorem A and Corollary B in [CDS19] for more precise statements.Moreover, Corollary 7.12 is consistent with the examples of complete expanding gradientK¨ahler-Ricci solitons discovered by Feldman, Ilmanen and Knopf [FIK03] on the total spaceof the tautological line bundles L − k , k > n , over CP n − . Indeed, these solutions on L − k are U ( n )-invariant and asymptotic to the cone C ( S n − / Z k ) endowed with the Euclideanmetric i∂∂ | · | , where Z k acts on C n diagonally. As noticed in [Sie13, Example 3 . . λ i ) ≤ i ≤ n of the Ricci tensor of these solitons are of the form: λ i = − ( k − n ) n e ( k − n ) φ − n e − φ , i = 1 , ..., n, λ n = ( k − n ) n e ( k − n ) φ − n e − φ ( φ + ( n − , where φ is a smooth positive function that behaves like | z | at infinity. Notice that this isconsistent with the (integral) Ricci curvature decay obtained in Theorem 7.1. Moreover, thescalar curvature of these metrics is equal to ( k − n ) n e ( k − n ) φ − n e − φ which is positive every-where on L − k . Finally, the trace at infinity of the Ricci tensor of these metrics is diagonal andis universally proportional to ( k − n ) n e k − n (cid:0) dr ⊗ dr + r η ⊗ η (cid:1) with the notations introducedabove. In particular, the transverse component of the trace at infinity of the Ricci tensorvanishes identically.8. A relative entropy for Ricci gradient expanders
We begin this section with a technical lemma that estimates various integrals involvingGaussian weights.
Lemma 8.1.
Let α > , β, γ ∈ R . There exists λ = λ ( α, β, γ ) > such that if r t ≥ λ , then ˆ t τ γ (cid:18) r τ (cid:19) β e − α r τ dτ ≤ C ( α, β ) t γ +1 (cid:18) r t (cid:19) β − e − α r t . Proof.
By a change of variable, one gets: ˆ t τ γ (cid:18) r τ (cid:19) β e − α r τ dτ = (cid:18) r (cid:19) γ ˆ t (cid:18) r τ (cid:19) β − γ e − α r τ dτ = (cid:18) r (cid:19) γ +1 ˆ + ∞ r t s β − γ − e − αs ds. RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 53
Now, by integrating by parts, ˆ + ∞ r t s β − γ − e − αs ds = 1 α (cid:18) r t (cid:19) β − γ − e − α r t + β − γ − α ˆ + ∞ r t s β − γ − e − αs ds ≤ α (cid:18) r t (cid:19) β − γ − e − α r t + 4 t | β − γ − | αr ˆ + ∞ r t s β − γ − e − αs ds. In particular, if we choose λ = λ ( α, β, γ ) > | β − γ − | ) · t ≤ αr , then oneabsorbs the last integral on the righthand side of the previous estimate to get the expectedresult. (cid:3) Let ( M ni , g i , ∇ g i f i ) i =1 , be two normalized asymptotically conical expanding gradient Riccisolitons coming out of the same cone ( C ( S ) , dr + r g S , r∂ r / t >
0, providing the limit exists: W rel ( g ( t ) , g ( t )) := lim R → + ∞ ˆ f ( t ) ≤ R e f ( t ) (4 πt ) n dµ g ( t ) − ˆ f ( t ) ≤ R e f ( t ) (4 πt ) n dµ g ( t ) ! . (8.1)The following theorem proves this quantity is well-defined. Theorem 8.2.
Let ( M ni , g i , ∇ g i f i ) i =1 , be two asymptotically conical gradient Ricci expanderscoming out of the same cone ( C ( S ) , dr + r g S , r∂ r / normalized as in Section 2.2. Thenthe limit in (8.1) exists and is finite for all time t > and is constant in time: −∞ < W rel ( g ( t ) , g ( t )) = W rel ( g ( s ) , g ( s )) < + ∞ , < s < t. Proof.
Let us prove that the relative entropy between ( g ( t )) t> and ( g ( t )) t> is independentof time provided it is well-defined.By using the change of variable y = ϕ it ( x ), ( ϕ it ) t> being the flow generated by −∇ g i f i /t , i = 1 ,
2, in the following integral shows immediately that ˆ f i ( t ) ≤ R e f i ( t ) (4 πt ) n dµ g i ( t ) = ˆ f i ≤ R e f i (4 π ) n dµ g i , which implies that W rel ( g ( t ) , g ( t )) does not depend on time provided it is well-defined.We decide to give another proof which is more in the spirit of the general case.We start by computing the evolution equation of the weights e f i ( t ) / (4 πt ) n/ where f i ( t ) :=( ϕ it ) ∗ f i . By using the soliton equations, see (2.3) – (2.5), (cid:0) ∂ t + ∆ g i ( t ) (cid:1) e f i ( t ) (4 πt ) n ! = (cid:16) ∂ t f i ( t ) − n t + ∆ g i ( t ) f i ( t ) + |∇ g i ( t ) f i ( t ) | g i ( t ) (cid:17) e f i ( t ) (4 πt ) n ! = R g i ( t ) e f i ( t ) (4 πt ) n . In particular, for a fixed positive radius R , ∂ t ˆ f i ( t ) ≤ R e f i ( t ) (4 πt ) n dµ g i ( t ) = ˆ f i ( t ) ≤ R ∂ t e f i ( t ) (4 πt ) n dµ g i ( t ) ! − ˆ f i ( t )= R ∂ t f i ( t ) |∇ g i ( t ) f i ( t ) | g i ( t ) e f i ( t ) (4 πt ) n ! dσ g i ( t ) = − ˆ f i ( t ) ≤ R ∆ g i ( t ) e f i ( t ) (4 πt ) n ! dµ g i ( t ) + ˆ f i ( t )= R |∇ g i ( t ) f i ( t ) | g i ( t ) e f i ( t ) (4 πt ) n ! dσ g i ( t ) = 0 , by Stokes formula, if we choose R sufficiently large so that { f i ( t ) = R } is a smooth hypersur-face.Therefore, by integrating with respect to time first and by letting R go to + ∞ gives theresult: W rel ( g ( t ) , g ( t )) = W rel ( g , g ), for all positive time t .Let us prove that W rel ( g ( t ) , g ( t )) is well-defined for all t >
0. Recall by (2.3) – (2.5) that if( ϕ it ) t> denotes the flow generated by −∇ g i f i /t such that ϕ it | t =1 = Id M and if f i ( t ) := ( ϕ it ) ∗ f i , ∂ t ( tf i ) = t R g i ( t ) , t > , or equivalently , ∂ t ( tf i ) = ∆ g i ( t ) ( tf i ) − n . (8.2)Let R > δ ∈ (0 , ϕ R,δ : [0 , + ∞ ) → [0 ,
1] be a smooth cut-off function such that ϕ R,δ ( r ) = 1 on [0 , R ], ϕ R,δ = 0 on [(1 + δ ) R, + ∞ ) and such that | ϕ ′ R,δ | ≤ c/ ( δR ) for somepositive universal constant c . We claim that the following limitlim R → + ∞ lim δ → ˆ M ϕ R,δ ( f ( t )) e f ( t ) (4 πt ) n dµ g ( t ) − ˆ M ϕ R,δ ( f ( t )) e f ( t ) (4 πt ) n dµ g ( t ) ! , (8.3)exists, is finite and is equal to W rel ( g ( t ) , g ( t )) for all t >
0. Observe that it suffices to provethis assertion at time t = 1 by considering parabolic rescalings of the solutions: g αi ( t ) := α − g i ( αt ), i = 1 , α > e f − f = e t ( f − f )( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =1 = 1 + ˆ ∂ τ e τ ( f − f )( τ ) dτ = 1 + ˆ ∂ τ ( τ ( f − f )) e τ ( f − f )( τ ) dτ = 1 + ˆ τ (R g ( τ ) − R g ( τ ) ) e τ ( f − f )( τ ) dτ = 1 + ˆ τ (R g ( τ ) − R g ( τ ) ) dτ + Q ( f − f ) , (8.4) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 55 where, Q ( f − f ) := ˆ τ (R g ( τ ) − R g ( τ ) ) (cid:16) e τ ( f − f )( τ ) − (cid:17) dτ. Here we have used (8.2) in the fourth line.Thanks to Theorem 7.1 and Corollary 7.11, we see that for any η >
0, we can estimate forany t ∈ (0 , t | R g ( t ) − R g ( t ) | ≤ C η e − (1 − η ) f ( t ) , on { f ( t ) ≥ R } , (8.5)where C is a time-independent positive constant. Observe that (8.5) is in particular true on { f ≥ R } ⊂ { f ( t ) ≥ R } .We can thus estimate the error term Q ( f − f ) outside K ⊂ M as follows: | Q ( f − f ) | ≤ C ˆ τ | R g ( τ ) − R g ( τ ) | | τ ( f − f )( τ ) | dτ ≤ ˆ τ | R g ( τ ) − R g ( τ ) | ˆ τ s (cid:12)(cid:12) R g ( s ) − R g ( s ) (cid:12)(cid:12) ds dτ ≤ C ˆ e − (1 − η ) f ( τ ) ˆ τ e − (1 − η ) f ( s ) ds dτ ≤ C ˆ e − − η ) f ( τ ) dτ ≤ Ce − − η ) f . Here, we have used repeatedly Lemma 8.1 in the fourth and the fifth lines and we have invoked(8.2) in the second line.Now, consider g σ := ( σ − · g + (2 − σ ) · g for σ ∈ [1 , g σ is a metric on M \ K . Then, (cid:18) dµ g dµ g (cid:19) = (cid:18) dµ g σ | σ =2 dµ g σ | σ =1 (cid:19) = 1 + tr g ( g − g ) − ˆ ˆ s | g − g | g σ dσds, where we have used the fact that if F ( σ ) := (cid:16) dµ gσ dµ g (cid:17) then d dσ F ( σ ) = ddσ tr g σ ( g − g ) = −| g − g | g σ . In particular, dµ g dµ g = 1 + 12 tr g ( g − g ) + Q ( g − g ) , (8.6)where we can estimate due to Theorem 7.1 and Corollary 7.11 that for η > | Q ( g − g ) | ≤ C η e − − η ) f . Next, observe that,tr g ( g − g ) = ˆ ∂ τ (cid:0) τ · tr g ( τ ) ( g ( τ ) − g ( τ )) (cid:1) dτ = ˆ τ (cid:0) tr g ( τ ) ( ∂ τ g − ∂ τ g ) (cid:1) dτ + ˆ (cid:0) ∂ τ g ∗ g ( τ ) − ∗ g ( τ ) − ∗ ( g − g )( τ ) (cid:1) dτ + ˆ tr g ( τ ) ( g ( τ ) − g ( τ )) dτ. (8.7)Note that by Theorem 2.1, we have that ∂ τ g i ( τ ) is uniformly bounded on { f i ( τ ) ≥ R } for all τ ∈ [0 ,
1] and i = 1 ,
2. Alternatively this follows from the quadratic curvature decay of g i (1).The last two integrals on the righthand side of (8.7) can then be estimated as follows: (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:0) ∂ τ g ∗ g ( τ ) − ∗ g ( τ ) − ∗ ( g − g )( τ ) (cid:1) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ | g ( τ ) − g ( τ ) | g ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) ˆ tr g ( τ ) ( g ( τ ) − g ( τ )) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ | g ( τ ) − g ( τ ) | g ( τ ) dτ . (8.8)Now, by the definition of a solution to the Ricci flow, one gets, ˆ τ tr g ( τ ) ( ∂ τ g ( τ ) − ∂ τ g ( τ )) dτ = − ˆ τ (cid:0) R g ( τ ) − R g ( τ ) (cid:1) dτ + ˆ g ( τ ) − ∗ ( g ( τ ) − g ( τ )) ∗ Ric( g ( τ )) dτ. (8.9)The second integral on the righthand side of (8.9) can be estimated as in (8.8): (cid:12)(cid:12)(cid:12)(cid:12) ˆ g ( τ ) − ∗ ( g ( τ ) − g ( τ )) ∗ Ric( g ( τ )) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ | g ( τ ) − g ( τ ) | g ( τ ) dτ. (8.10)To sum it up, (8.4), (8.6) together with (8.7), (8.8) and (8.10) lead to: (cid:12)(cid:12)(cid:12)(cid:12) e f − f dµ g dµ g − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ | g ( τ ) − g ( τ ) | g ( τ ) dτ + C η e − − η ) f . (8.11)As an intermediate step, notice that: ϕ R,δ ( f ) e f − f dµ g dµ g − ϕ R,δ ( f ) = ϕ R,δ ( f ) (cid:18) e f − f dµ g dµ g − (cid:19) + ( ϕ R,δ ( f ) − ϕ R,δ ( f ))= ϕ R,δ ( f ) (cid:18) e f − f dµ g dµ g − (cid:19) + (cid:18) ˆ ϕ ′ R,δ ( f σ ) dσ (cid:19) ( f − f ) (8.12)where f σ = ( σ − f + (2 − σ ) f for σ ∈ [1 , RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 57
Claim 8.3.
For
R > R , it holds lim sup δ → ˆ ˆ E √ R ϕ R,δ ( f ) | g ( τ ) − g ( τ ) | g ( τ ) e f dµ g dτ ≤ CR − . Proof of Claim 8.3.
For τ >
0, define h ( τ ) := g ( τ ) − g ( τ ) and correspondingly ˆ h ( τ ) := f n ( τ ) e f ( τ ) h ( τ ) . Then, by the Cauchy-Schwarz inequality, for
R > R , ˆ ˆ E √ R ϕ R,δ ( f ) | g ( τ ) − g ( τ ) | g ( τ ) e f dµ g dτ ≤ ˆ ˆ E √ R | ˆ h ( τ ) | g ( τ ) e f − f ( τ ) f − n ( τ ) dµ g dτ ≤ ˆ (cid:18) ˆ E √ R | ˆ h ( τ ) | g ( τ ) f − n +12 ( τ ) dµ g (cid:19) (cid:18) ˆ E √ R e f − f ( τ )) f − n − ( τ ) dµ g (cid:19) dτ. Now, since g and g ( τ ) are uniformly-in-time equivalent on E √ R by Theorem 2.1, Theorem7.1 and the change of variable theorem lead to: ˆ E √ R | ˆ h ( τ ) | g ( τ ) f − n +12 ( τ ) dµ g ≤ C ˆ E √ R | ˆ h ( τ ) | g ( τ ) f − n +12 ( τ ) dµ g ( τ ) = Cτ n ˆ ϕ τ ( E √ R ) | ˆ h | g f − n +12 dµ g ≤ Cτ n ˆ E √ R √ τ | ˆ h | g f − n +12 dµ g ≤ C τ n +12 √ R . Finally, by invoking Fubini’s theorem and Lemma 8.1, ˆ ˆ E √ R e f − f ( τ )) f − n − ( τ ) dµ g dτ = ˆ E √ R (cid:18) ˆ e − f ( τ ) f − n − ( τ ) dτ (cid:19) e f dµ g ≤ C ˆ E √ R f − n +12 dµ g ≤ C ( R ) < + ∞ . This ends the proof of Claim 8.3. (cid:3)
The following claim takes care of the second term on the righthand side of (8.12):
Claim 8.4. lim R → + ∞ lim δ → ˆ M ˆ ϕ ′ R,δ ( f σ ) ( f − f ) dσe f dµ g , exists and is finite.Proof of Claim 8.4. For reasons that will be made be clear below, it is more convenient toexpress the difference f − f in terms of the Morse flow ( ψ στ ) τ> associated to each function r σ := 2 √ f σ , σ ∈ [1 ,
2] instead of the flow generated by −∇ g i f i /τ , i = 1 , σ ∈ [1 , ∂ τ ψ στ = − τ − ∇ g σ f σ |∇ g σ r σ | g σ ◦ ψ στ =: − τ − X σ ◦ ψ στ , τ > , ψ στ =1 = Id M . (8.13)The advantage of this flow lies in the following basic property on the action of this one-parameter family of diffeomorphisms on the level sets of r σ : by using the very definition ofthe flow given in 8.13, one gets for τ ∈ (0 ,
1] and ρ > ψ στ (cid:0) S σρ (cid:1) = S σ ρ √ τ , where S σρ := { r σ = ρ } .Moreover, observe that [(2.16), Lemma 2.4] and (2.8) give: | X σ − X | g σ ≤ C (cid:0) r σ | h | g σ + r − σ (cid:1) ≤ C r − σ , where X := ∇ g f = ∇ g f and C is a positive constant uniform in σ ∈ [1 ,
2] that may varyfrom line to line. Here we have used Theorem 2.1 ensuring that | h | g σ = O ( r − σ ) in the secondinequality.By mimicking the Taylor expansions done in (8.4) and (8.7)–(8.10), one gets on the onehand, f − f = ˆ ∂ τ ( τ ( ψ στ ) ∗ ( f − f )) dτ = ˆ ( ψ στ ) ∗ (( f − f ) + X σ · ( f − f )) dτ = ˆ ( ψ στ ) ∗ (cid:0) ( f − |∇ g f | g − ( f − |∇ g f | g ) (cid:1) dτ + ˆ ( ψ στ ) ∗ (( X σ − ∇ g f ) · f − ( X σ − ∇ g f ) · f ) dτ = ˆ ( ψ στ ) ∗ (R g − R g ) dτ + ˆ ( ψ στ ) ∗ (( X σ − X ) · ( f − f )) dτ, (8.14)where we have used (2.5) in the last line.Now on the other hand, observe that:( X σ − X ) · ( f − f ) = 1 |∇ g σ r σ | g σ ( ∇ g σ f σ − X ) · ( f − f ) + (cid:18) |∇ g σ r σ | g σ − (cid:19) ( g − g )( X, X ) . Thanks to [(2.16), Lemma 2.4] and (2.8), there exists a positive constant C independent of σ ∈ [1 ,
2] that may vary from line to line such that: | ( X σ − X ) · ( f − f ) | ≤ C r σ | h | g σ |∇ g σ ( f − f ) | g σ + C r − σ | h | g σ | X | g σ ≤ C r σ | h | g σ + C r − σ | h | g σ ≤ C r − σ | h | g σ , (8.15)where we have used [(2.16), Lemma 2.4] once more in the second line together with Theorem2.1 ensuring that | h | g σ = O ( r − σ ). RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 59
It remains to relate the difference of the scalar curvatures on the righthand side of (8.14)to the difference of metrics h as we did in estimates (8.7)–(8.10). To do so, define the one-parameter family of metrics g σi ( τ ) := τ ( ψ στ ) ∗ g i for i = 1 , σ ∈ [1 ,
2] and τ > ∂ τ g σi = g σi ( τ ) τ − τ − L ( ψ στ ) ∗ X σ ( g σi ( τ ))= − g σi ( τ )) − τ − L ( ψ στ ) ∗ ( X σ − X ) ( g σi ( τ )) , (8.16)where we have used the soliton equation (2.2) in the last line. In particular, by tracing (8.16)for i = 1 , τ tr g σi ( τ ) ( ∂ τ g σi ) = − τ R g σi ( τ ) − g σi ( τ ) (( ψ στ ) ∗ ( X σ − X )) . (8.17)Based on (8.17), the same reasoning that led to estimates (8.7)–(8.10) gives: (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( ψ στ ) ∗ (R g − R g + div g ( X σ − X ) − div g ( X σ − X )) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | tr g h | + C ˆ | g σ ( τ ) − g σ ( τ ) | g σ ( τ ) dτ. (8.18)As for the last term on the lefthand side of (8.18), we linearize carefully the divergence of avector field V as follows:div g V = div g V + 12 V · tr g h + O ( | V | g | h | g ) . (8.19)This essentially comes from [CLN06, Section 5, Chapter 2].Applying (8.19) to V := X σ − X together with [(2.16), Lemma 2.4], as well as Corollary7.11 to estimate ∇ g σ tr g h , leads to an improvement of (8.18): (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( ψ στ ) ∗ (cid:18) R g − R g + 12 (cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ f σ · tr g h (cid:19) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | tr g h | + C ˆ | g σ ( τ ) − g σ ( τ ) | g σ ( τ ) dτ. (8.20)Combining estimates (8.14), (8.15) and (8.20) finally leads to: (cid:12)(cid:12)(cid:12)(cid:12) ˆ M ˆ ϕ ′ R,δ ( f σ ) ( f − f ) dσe f dµ g − ˆ M ˆ ϕ ′ R,δ ( f σ ) ˆ ( ψ στ ) ∗ (cid:20)(cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ f σ · tr g h (cid:21) dτ dσe f dµ g (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ M ˆ ˆ | ϕ ′ R,δ ( f σ ) | (cid:16) | h | g σ + | h σ ( τ ) | g σ ( τ ) (cid:17) dτ dσe f dµ g , (8.21)for some positive constant C uniform in σ ∈ [1 ,
2] where h σ ( τ ) := g σ ( τ ) − g σ ( τ ) . In order to prove Claim 8.4, we first prove that the righthand side of (8.21) converges to 0as δ tends to 0 and R goes to + ∞ . By Cauchy-Schwarz inequality and Theorem 7.1 applied to each metric g σ and potentialfunction f σ (see Remark 7.5), observe that: ˆ M | ϕ ′ R,δ ( f σ ) | | h | g σ e f σ dµ g σ ≤ CRδ ˆ R ≤ f σ ≤ R (1+ δ ) | ˆ h | g σ f − n σ dµ g σ ≤ CRδ ˆ √ R (1+ δ )2 √ R ρ − n ˆ S σρ | ˆ h | g σ dρ ≤ CRδ ˆ √ R (1+ δ )2 √ R ρ − n +12 ˆ S σρ | ˆ h | g σ ! dρ ≤ CRδ ˆ √ R (1+ δ )2 √ R ρ − dρ ≤ CR log (1 + δ ) δ (8.22)where C is a uniform positive constant which does not depend on σ ∈ [1 , δ ∈ (0 ,
1] and R ≥ R > R → + ∞ lim sup δ → ˆ ˆ M ˆ | ϕ ′ R,δ ( f σ ) | | h σ ( τ ) | g σ ( τ ) dτ e f σ dµ g σ dσ = 0 . (8.23)We proceed analogously as before by defining f σ ( τ ) := ( ψ στ ) ∗ f σ and ˆ h σ ( τ ) := f n σ ( τ ) e f σ ( τ ) h σ ( τ )and by observing that: ˆ M | ϕ ′ R,δ ( f σ ) | ˆ | h σ ( τ ) | g ( τ ) dτ e f σ dµ g σ ≤≤ CRδ ˆ ˆ R ≤ f σ ≤ R (1+ δ ) | ˆ h σ ( τ ) | g ( τ ) e f σ − f σ ( τ ) f − n σ ( τ ) dµ g σ dτ ≤ CRδ ˆ ˆ R ≤ f σ ≤ R (1+ δ ) | ˆ h σ ( τ ) | g ( τ ) f − n σ ( τ ) dµ g σ ( τ ) dτ ≤ CRδ ˆ ˆ Rτ ≤ f σ ≤ R (1+ δ ) τ | ˆ h | g f − n σ dµ g σ τ n dτ, where we have used the fact that f σ ≤ f σ ( τ ) for τ ∈ (0 ,
1] and a change of variable in the lastline. We notice that the use of the Morse flow ( ψ στ ) τ> is crucially entering the proof of thisclaim here.Now, reasoning as in (8.22), ˆ M | ϕ ′ R,δ ( f σ ) | ˆ | h σ ( τ ) | g ( τ ) dτ e f σ dµ g σ ≤ CRδ ˆ ˆ √ Rτ − (1+ δ )2 √ Rτ − ρ − dρdτ ≤ CR log (1 + δ ) δ . This ends the proof of (8.23).We are left with proving that:lim R → + ∞ lim δ → ˆ M ˆ ϕ ′ R,δ ( f σ ) ˆ ( ψ στ ) ∗ (cid:20)(cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ f σ · tr g h (cid:21) dτ dσe f dµ g , (8.24) RELATIVE ENTROPY AND A UNIQUE CONTINUATION RESULT FOR RICCI EXPANDERS 61 exists and is finite.Since | g σ − g | g ≤ | h | g and tr g σ h = tr g h + g − σ ∗ g − ∗ h ∗ h poinwise, as well as Corollary7.11 to estimate ∇ g σ tr g h and arguing as in the previous steps, it is equivalent to provethat the previous limit exists and is finite with respect to the measure dµ g σ instead and theterm tr g σ h in place of tr g h . Due to Fubini’s theorem together with the change of variabletheorem, observe that: ˆ I R,δ ( τ ) dτ := ˆ ˆ M ϕ ′ R,δ ( f σ )( ψ στ ) ∗ (cid:20)(cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ f σ · tr g σ h (cid:21) e f dµ g σ dτ = ˆ ˆ M ϕ ′ R,δ ( τ f σ ) (cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ f σ · tr g σ h e ( ψ στ − ) ∗ f dµ ( ψ στ − ) ∗ g σ dτ = ˆ ˆ M (cid:18) |∇ g σ r σ | g σ − (cid:19) ∇ g σ ( ϕ R,δ ( τ f σ )) · tr g σ h e ( ψ στ − ) ∗ f dµ ( ψ στ − ) ∗ g σ τ − dτ. (8.25)Now, introduce the following auxiliary weight for δ > R sufficiently large and τ ∈ (0 , w σ ( τ ) := (cid:18) |∇ g σ r σ | g σ − (cid:19) e ( ψ στ − ) ∗ f dµ ( ψ στ − ) ∗ g σ dµ g σ , whose definition is motivated by the computation (8.25). Notice that (2.8), [(2.12), Lemma2.4] and the definition of the Morse flow imply that on { f σ ≥ τ − R } : | w σ ( τ ) | ≤ Cτ n f − σ e τf σ , (8.26)where C is a positive constant uniform in τ ∈ (0 ,
1] and σ ∈ [1 , τ ∈ (0 , I R,δ ( τ ) = − ˆ M ϕ R,δ ( τ f σ ) div g σ ( w σ ( τ ) ∇ g σ tr g σ h ) dµ g σ . By using the very definition of ˆ h , and by invoking the same reasoning that led to the proofof (8.23), showing (8.24) amounts to showing that the limit of ˆ J R,δ ( τ ) := − ˆ ˆ M ϕ R,δ ( τ f σ ) div g σ (cid:16) w σ ( τ ) f − n σ e − f σ ∇ g σ tr g σ ˆ h (cid:17) dµ g σ τ − dτ, as δ tends to 0 and as R goes to + ∞ exists and is finite.To do so, we invoke Corollary 2.7 to get:div g σ (cid:16) w σ ( τ ) f − n σ e − f σ ∇ g σ tr g σ ˆ h (cid:17) == w σ ( τ ) f − n σ e − f σ L − n tr g σ ˆ h + f − n σ e − f σ ∇ g σ w σ ( τ ) · tr g σ ˆ h = 2 n +1 w σ ( τ ) div g σ B + n w σ ( τ ) f − n +22 σ e − f σ ∇ g σ ∇ gσ f σ tr g σ ˆ h + w σ ( τ ) f − n σ e − f σ tr g σ R [ˆ h ] + f − n σ e − f σ ∇ g σ w σ ( τ ) · tr g σ ˆ h. (8.27) Thanks to Theorem 7.1 and estimate (8.26), ˆ ˆ M ϕ R,δ ( τ f σ ) | w σ ( τ ) | h f − σ (cid:12)(cid:12)(cid:12) ∇ g σ ∇ gσ f σ tr g σ ˆ h (cid:12)(cid:12)(cid:12) + | tr g σ R [ˆ h ] | i f − n σ e − f σ dµ g σ τ − dτ ≤ C ˆ ˆ M ϕ R,δ ( τ f σ ) f − − n σ e ( τ − f σ h f − σ (cid:12)(cid:12)(cid:12) ∇ g σ ∇ gσ f σ ˆ h (cid:12)(cid:12)(cid:12) + f − σ (cid:16) | ˆ h | g σ + |∇ g σ ˆ h | g σ (cid:17)i dµ g σ τ n − dτ ≤ C ˆ ˆ M f − − n σ h f − σ (cid:12)(cid:12)(cid:12) ∇ g σ ∇ gσ f σ ˆ h (cid:12)(cid:12)(cid:12) + f − σ (cid:16) | ˆ h | g σ + |∇ g σ ˆ h | g σ (cid:17)i dµ g σ τ n − dτ < + ∞ . As for the term involving the Bianchi one-form B , we integrate by parts to get: ˆ M ϕ R,δ ( τ f σ ) w σ ( τ ) div g σ B dµ g σ = − τ ˆ M ϕ ′ R,δ ( τ f σ ) w σ ( τ ) g σ ( ∇ g σ f σ , B ) dµ g σ − ˆ M ϕ R,δ ( τ f σ ) g σ ( ∇ g σ w σ ( τ ) , B ) dµ g σ . (8.28)Observe that the first integral on the righthand side of (8.28) can be estimated as follows for τ ∈ (0 , τ (cid:12)(cid:12)(cid:12)(cid:12) ˆ M ϕ ′ R,δ ( τ f σ ) w σ ( τ ) g σ ( ∇ g σ f σ , B ) dµ g σ (cid:12)(cid:12)(cid:12)(cid:12) ≤ CRδ τ n +1 ˆ Rτ ≤ f σ ≤ R (1+ δ ) τ f − σ e τf σ |B| g σ dµ g σ ≤ CRδ τ n +1 ˆ Rτ ≤ f σ ≤ R (1+ δ ) τ f − σ e f σ |B| g σ dµ g σ . By reasoning as in (8.22) based on the decay of the frequency function associated to B estab-lished in Proposition 5.4, one gets:lim R → + ∞ lim sup δ → ˆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ M ϕ ′ R,δ ( τ f σ ) w σ ( τ ) g σ ( ∇ g σ f σ , B ) dµ g σ (cid:12)(cid:12)(cid:12)(cid:12) dτ = 0 . It remains to estimate the gradient of w σ ( τ ) in order to handle the remaining terms on therighthand side of (8.27) and (8.28). By using that the gradient of f σ grows linearly and theuniform equivalence of the metrics g σ ( τ ) for τ ∈ (0 ,
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Alix Deruelle: Institut de Math´ematiques de Jussieu, Paris Rive Gauche (IMJ-PRG) UPMC -Campus Jussieu, 4, place Jussieu Boite Courrier 247 - 75252 Paris Cedex 05, France
Email address : [email protected] Felix Schulze: Mathematics Institute, Zeeman Building, University of Warwick, CoventryCV4 7AL, UK
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