Numerical Bifurcation Analysis of the Conformal Method
NNUMERICAL BIFURCATION ANALYSIS OF THE CONFORMAL METHOD
JAMES DILTS, MICHAEL HOLST, TAMARA KOZAREVA, AND DAVID MAXWELLA
BSTRACT . The conformal formulation of the Einstein constraint equations has beenstudied intensively since the modern version of the conformal method was first pub-lished in the early 1970s. Proofs of existence and uniqueness of solutions were limitedto the constant mean curvature (CMC) case through the early 90s, with analogous resultsfor the near-CMC case beginning to appear thereafter. In the last decade, there has beensome limited progress towards understanding the properties of the conformal methodfor far-from-CMC solutions as well. Although it was initially conceivable that that thesefar-from-CMC results would lead to a solution theory for the non-CMC case that wouldmirror the good properties of the CMC and near-CMC cases, examples of bifurcationsand of nonexistence of solutions have been since discovered. Nevertheless, the generalproperties of the conformal method for far-from-CMC data remain unknown. In thisarticle we apply analytic and numerical continuation techniques to the study of the con-formal method, in an attempt to give some insight into what the solution behavior is inthe far-from-CMC case in various scenarios. C ONTENTS
1. Introduction 11.1. The Conformal Method 21.2. Far-from-CMC results for the conformal method 42. Tools from Bifurcation Analysis 62.1. Analytic Bifurcation Theory 62.2. Numerical Bifurcation Analysis 73. Numerical Results 83.1. Sign-changing mean curvature 83.2. Constant-sign mean curvatures 134. Discussion 225. Conclusion 246. Acknowledgments 24Appendix A. A constant norm TT tensor on S × S NTRODUCTION
In general relativity spacetime is described by by a Lorentzian manifold ( M , g ) , thatis, a four-dimensional differentiable manifold M endowed with a non-degenerate, sym-metric rank (0 , tensor field g on M whose signature is ( − , , , . The space-time Date : September 17, 2018.2010
Mathematics Subject Classification.
Key words and phrases. nonlinear partial differential equations, the conformal method, folds, bifurca-tion, pseudo-arclength continuation, AUTO.JD was supported in part by NSF DMS/RTG Award 1345013 and DMS/FRG Award 1262982.MH was supported in part by NSF DMS/FRG Award 1262982 and NSF DMS/CM Award 1620366.TK was supported in part by NSF DMS/RTG Award 1262982.DM was supported in part by NSF DMS/FRG Award 1263544. a r X i v : . [ g r- q c ] A ug M , g ) is required to satisfy the Einstein field equations , Ric g − R g = 8 πGc T , (1.1)where
Ric g is the Ricci curvature tensor, R g its scalar ( R := Ric ab g ab ), and T is thestress energy-momentum tensor of any matter fields present. Once a time function hasbeen chosen, space-time is foliated by space-like constant-time hypersurfaces Σ t andevolution and constraint equations are obtained by considering the projections of thefield equations (1.1) in directions tangent and orthogonal to the space-like hypersurfaces.The evolution equations can be cast as a first-order system for the first and second fun-damental forms associated with the time slices, namely the three-metric ˆ g and extrinsiccurvature ˆ k . With ˆ g and ˆ k symmetric tensors, this represents 12 equations for the 12components of ˆ g and ˆ k , with the equations being first-order in time and second-order inspace.The four constraint equations on the 12 degrees of freedom are R g + τ − | ˆ k | g = 0 , (1.2) div ˆ k − dτ = 0 (1.3)with R g the scalar curvature of g and τ = ˆ g ij ˆ k ij the trace of the extrinsic curvature. Theseequations are direct consequences of the Gauss-Codazzi-Mainardi conditions which arerequired for an -manifold to arise as a submanifold of a -manifold. If matter and/orenergy sources are present, then the 12 evolution equations and the four constraint equa-tions (1.2)–(1.3) contain additional terms. The constraint equations are obviously under-determined as a stand-alone system of equations for the initial data, in that they fix onlysome part of the 12 degrees of freedom. One must therefore make a choice of whichparts of the initial data one wishes to fix, and which parts are to be determined by theconstraint equations (1.2)–(1.3). The conformal method , described in the next section,is an approach to parameterizing the initial data so that the constraint equations for theremaining degrees of freedom can potentially be uniquely solved. It provides an effec-tive parameterization of the constant-mean-curvature (CMC) solutions of the constraintequations, and is generally effective for near-constant mean curvatures as well. There hasbeen recent progress in determining its properties in the far-from-CMC setting, and whatlittle we know indicates the situation is somewhat complex. The aim of this paper is tobring numerical methods, and numerical bifurcation theory specifically, to yield furtherinsight into what can be expected for the conformal method when applied to far-from-CMC initial data.1.1. The Conformal Method.
The conformal method was proposed by Lichnerowiczin 1944 [37], and then substantially generalized in the 1970s by York [49], among otherauthors. The method is based on a splitting of the initial data ˆ g (a Riemannian metric ona space-like hypersurface Σ t ) and ˆ k (the extrinsic curvature of the hypersurface Σ t ) intoeight freely specifiable pieces, with four remaining pieces to be determined by solvingthe four constraint equations.The pieces of the initial data that are specified as part of the method are called the seeddata and are comprised of a spatial background metric g on Σ t , defined up to multiplica-tion by a conformal factor (five free functions), a positive function N (a so-called densi-tized lapse), a function τ , and a transverse, traceless (TT) tensor σ ij (effectively two freefunctions, as it is symmetric, trace-free and divergence free). The two remaining piecesof the initial data to be determined by the constraints are a scalar conformal factor ϕ > nd a vector potential W . The full spatial metric ˆ g and the extrinsic curvature ˆ k are thenrecovered from ϕ , W , and the eight specified functions from the expressions: ˆ g = ϕ g, and ˆ k = ϕ − (cid:2) σ + N ( ckW ) (cid:3) + ϕ τ g. This transformation has been engineered so thatthe constraints (1.2)–(1.3) reduce to coupled PDEs for ϕ and W with standard ellipticoperators as their principle parts; in three spatial dimensions the equations are − ϕ + Rϕ + 23 τ ϕ − (cid:12)(cid:12)(cid:12)(cid:12) σ + 12 N L W (cid:12)(cid:12)(cid:12)(cid:12) ϕ − = 0 , (1.4) − div (cid:20) N L W (cid:21) + 23 ϕ d τ = 0 . (1.5)Here, ∆ is the Laplace-Beltrami operator with respect to the background metric g , L denotes the conformal Killing operator ( L W ) ij = ∇ i W j + ∇ j W i − ( ∇ k W k ) g ij , and τ = ˆ k ij ˆ g ij is again the trace of the extrinsic curvature. We note that the densitized lapse N is more commonly associated with the conformal thin-sandwich method[50], but theequivalence of that method with the standard conformal method was demonstrated in[43]. In particular, the conformal method represents a family of parameterizations of theconstraint equations within a given conformal class of metric, one for each choice of N (or one for each choice of metric representing the conformal class). A detailed overviewof the conformal method, and its variations, may be found in the 2004 survey [3].When τ is constant (i.e. when the Cauchy surface Σ t has constant mean curvature),then the term in equation (1.5) involving d τ vanishes, and the two equations decouple.The only solutions of (1.5) have L W = 0 , and it remains only to solve the Lichnerowiczequation (1.4) with L W = 0 ; a similar decoupling occurs for non-vacuum seed datadata. Initial work starting with [47] focused on the CMC case of the conformal method,and a full description of the parameterization on compact manifolds was achieved in[31]. The theory depends on on the Yamabe invariant Y ( g ) of the seed metric , and issummarized in Table 1.T ABLE
1. Constant mean curvature (CMC) solvability [31] τ = 0 , σ ≡ τ = 0 , σ (cid:54)≡ τ (cid:54) = 0 , σ ≡ τ (cid:54) = 0 , σ (cid:54)≡ Y ( g ) > None Unique None Unique Y ( g ) = 0 Unique up to homotopy None None Unique Y ( g ) < None None Unique UniqueThe CMC conformal method is also well understood in other asymptotic geometries(e.g, asymptotically Euclidean [9][39], asymptotically hyperbolic [2]). It has been usedin a number of applications, including results for open manifolds with interior “blackhole” boundary models [18, 39], results allowing for “rough” data [40, 7], and numericalrelativity (e.g. [15, 14])Investigations of near-CMC seed data began to appear in the mid-90s, and we point to[32], [1] and [33] which developed the near-CMC theory on compact manifolds sum-marized in Table 2. Recall that Y ( g ) > if and only if g has a conformally related metric with positive scalar curvature,and similarly for Y ( g ) = 0 and Y ( g ) < . The specific conditions characterizing near-CMC seed data depend on the context but all involve con-trol on dτ /τ . ABLE
2. Near-CMC Solvability [32, 1, 33] τ (cid:54)≡ , σ ≡ τ (cid:54)≡ , σ (cid:54)≡ Y ( g ) > None Unique Y ( g ) = 0 None Unique Y ( g ) < Unique UniqueThe existence results in this table require an additional hypothesis that the backgroundmetric not have any conformal Killing fields, and it has been recently shown [25] thatin some cases this hypothesis is necessary. Conformal Killing fields form the kernelof the self-adjoint elliptic operator appearing in equation (1.5), and their presence in-terferes with iterative approaches to obtaining solutions; nearly all theorems concerningnon-CMC seed data for the conformal method assume there are no conformal Killingfields. Extensions of the near-CMC theory are available in other asymptotic geometries(asymptotically Euclidean [9], asymptotically hyperbolic [34]), and it has been appliedin numerical relativity [16, 5, 24, 48].1.2.
Far-from-CMC results for the conformal method.
In the last decade, a handfulof results have appeared concerning the conformal method in the far-from CMC setting.They provide the main context needed to understand our numerical experiments, and wesummarize them in somewhat more detail in this section.Building off of a strictly non-vacuum result [29], the following theorem provides ex-istence, in vacuum, for arbitrary mean curvatures, so long as the background metric isYamabe positive and the TT tensor is sufficiently small.
Theorem 1.1 ([41]) . Let ( M, g ) be a compact Yamabe-positive manifold with no confor-mal Killing fields. Given arbitrary vacuum seed data ( τ, σ, N ) , if σ (cid:54)≡ is sufficientlysmall (with smallness depends on the choice of τ ), then there exists at least one solutionof the conformally parameterized constraint equations (1.4) - (1.5) . Variations on this theorem have subsequently been demonstrated in other contexts(asymptotically Euclidean manifolds [20, 27, 4], manifolds with asymptotically cylindri-cal or periodic ends [12, 13], and other settings [30, 28, 19, 29, 4]). Although Theorem1.1 is silent on the issue of uniqueness, it is consistent with Table 2 extending generallyfor arbitrary mean curvatures, and there was some optimism that this might be the casewhen [29, 41] appeared. Although it is evidently a far-from CMC result, the alterna-tive perspective of [22] demonstrates that the solutions found in Theorem 1.1 can alsobe thought of as rescalings of near-CMC solutions that are perturbations off of τ ≡ solutions, as allowed in Table 1 for Yamabe-positive seed data.The following result is a consequence of a groundbreaking blowup analysis for theconformal method. Theorem 1.2 ([17]) . Let ( M n , g ) be a compact manifold without conformal Killingfields, and let ( σ, τ, N ) be vacuum seed data on it with τ (cid:54) = 0 having constant sign.If there does not exist a solution of the conformally parameterized constraint equations (1.4) - (1.5) , then there exists a solution of the limit equation div (cid:20) N L W (cid:21) = α (cid:114) n − n (cid:12)(cid:12)(cid:12)(cid:12) N L W (cid:12)(cid:12)(cid:12)(cid:12) dττ . (1.6) for some α ∈ (0 , . ee also [23, 21], where Theorem 1.2 has been extended to other settings. The mainidea behind the proof of the theorem is that if one cannot maintain L ∞ control on approx-imate solutions φ of the conformally parameterized constraint equations, then rescalingsof the approximations eventually lead to a solution of the blowup profile (1.6). One po-tential application of Theorem 1.2 is to show solutions exist by ruling out the possibilityof solutions of the limit equation (1.6), and indeed [17] contains a near-CMC existencetheorem with a large perturbation constant based on this idea. Although Theorem 1.2has proved difficult to apply in practice, because of the challenge of working with equa-tion (1.6), our numerical work suggests that it plays a decisive role in analyzing system(1.4)-(1.5) for constant sign mean curvatures.For mean curvatures that change sign, little is known in general aside from Theorem1.1. However, [42] contains an analysis of some very specific families of seed data onthe flat torus that includes the sign changing case. Among the seed data considered thereis a family of mean curvatures of the form τ = 1 + aξ where ξ is a particular dicontinuous, piecewise constant function equal to ± and TTtensors of the form η ˆ σ for a particular reference TT tensor ˆ σ and an arbitrary constant η . Theorem 1.3 ([42]) . For particular seed data on the flat torus T of the form ( τ = 1 + aξ, σ = η ˆ σ, N ) , if | a | > (and hence τ changes sign), there is an η ∗ > depending on a so that if η > η ∗ there is no solution of system (1.4) - (1.5) with the symmetry of the data, but if < η < η ∗ then there are at least two solutions. This was the first theorem to demonstrate the existence of multiple solutions of thevacuum conformal method in the far-from-CMC setting. Although in involves Yamabe-null examples, it cast doubt on the possibility that Theorem 1.1 concerning Yamabe-positive seed data could be extended to include a uniqueness statement or that its small-TT tensor requirement could be dropped. The follow-up study [44] contains additionalresults on related families of far-from-CMC data.Using ideas from [17], Nguyen [45] recently showed conclusively that the restrictionsof Theorem 1.1 are essential.
Theorem 1.4 ([45]) . Let ( M, g ) be a compact Yamabe-positive manifold ( M, g ) with noconformal Killing fields. Consider a family of seed data ( τ = ξ a , µσ, N ) with a > , µ ∈ R , where ξ is a fixed positive function. Assume additionally ξ satisfies (cid:12)(cid:12)(cid:12)(cid:12) L (cid:18) dξξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:12)(cid:12) dξξ (cid:12)(cid:12)(cid:12)(cid:12) (1.7) for some c > , and that σ is supported away from the critical points of τ . Then if a issufficiently large, and if | µ | is larger than a threshold depending on a , the conformallyparameterized constraint equations (1.4) - (1.5) do not admit a solution. For the same a ,there is a sequence µ k → such that there are at least two solutions to these equationsalong the sequence, and such that there is a solution with µ = 0 .The set of seed data satisfying these conditions is nonempty. The restrictions on the seed data in Theorem 1.4 are quite severe, but the result isremarkable nevertheless. In particular, the existence of solutions at σ ≡ for Yamabe-positive data was a surprise. In addition to proving Theorem 1.4, [45] gives insight intothe role of the deficiency parameter α in the limit equation (1.6) and these unexpected ≡ solutions. On a Yamabe-positive manifold, if there does not exist a solution forgiven seed data, and if there does not exist a solution of the limit equation with α = 1 ,then there is, in fact, a solution for the same seed data but with σ ≡ .Very recently, Nguyen obtained the following extension of Theorem 1.4. Theorem 1.5 ([46]) . Let ( M, g ) be a compact Yamabe-positive manifold ( M, g ) with noconformal Killing fields. Consider a family of seed data ( τ = ξ a , µσ, N ) with a > , µ ∈ R , and where ξ is a fixed positive function satisfying inequality (1.7) . Then, for each a sufficiently large, there is a m depending on a such that if < µ < m there areat least two solutions of the conformally parameterized constraint equations, and when µ = 0 there is at least one. While Theorem 1.5 significantly relaxes many of the hypotheses of Theorem 1.4 andstrengthens some of its conclusions, it makes no claims concerning non-existence forlarge µ , a point we will revisit in our numerical experiments. Inequality (1.7) neededfor Theorems 1.4 and 1.5 is non-generic, and our numerical work gives insight about theextent to which Theorem 1.5 holds more generally.In summary, we have the following. • On Yamabe-positive manifolds, arbitrary mean curvatures can be used for seeddata, so long as the TT tensor is small enough. For certain families of Yamabe-positive seed data, there are multiple solutions for small TT tensors. Moreover,for these families of seed data, there exist solutions at σ ≡ , something ruledout in the near-CMC case (Table 2). Additionally, there are some cases wherethere are no solutions for large TT tensors. • On a particular Yamabe-null manifold, with particular sign-changing seed data,we have multiple solutions for small TT tensors, and nonexistence (within thesymmetry class) for large TT tensors. • Nothing specific is known for Yamabe-negative seed data. • The limit equation criterion holds for all Yamabe classes, but the question of theexistence of solutions of the limit equation is essentially open.These limited results provide our motivation to look at analytic bifurcation theory andclosely related numerical continuation methods to try to gain intuition for what can beexpected more generally from the conformal method in the far-from CMC regime.2. T
OOLS FROM B IFURCATION A NALYSIS
The unexpectedly complex behavior of solutions to the conformal method equationsin the far-from-CMC regime leads one to the language and technical tools of analyticbifurcation theory . This area of nonlinear analysis is the study of the branching of solu-tions of nonlinear problems with respect to the parameters.
Numerical continuation (or numerical homotopy methods ) is a related area and refers to a collection of practical nu-merical methods for computing solution branches of nonlinear problems through criticalpoints such as folds and bifurcations.2.1.
Analytic Bifurcation Theory.
To explain the main ideas that are relevant here,consider again the PDE representation of the conformal method (1.4)–(1.5), but writtenmore simply as the abstract nonlinear problem: Find u ∈ X such that F ( u, λ ) = 0 , (2.1)where F : X × Z → Y for suitably chosen Banach spaces X , Y , and Z , and where λ ∈ Z represents the parameters of interest that are moved through the parameter space . In the setting of the conformal method and its variations, one can consider variousparameterizations. Once a parameterization λ is chosen, one is interested in the localbehavior of the solution curve u ( λ ) in a neighborhood of a known solution u ( λ ) . Thetechniques of both analytic and numerical bifurcation analysis rely on the Implicit Func-tion Theorem (IFT) as the basic tool for doing this exploration.
Given F : X × Z → Y , where X , Y , and Z are Banach spaces, if F ( u , λ ) = 0 , if F and F u (the Frech´et derivative of F ) are continuous on some region U × V ⊂ X × R containing ( u , λ ) , and if F u ( u , λ ) is nonsingular with a bounded inverse, then thereis a unique branch of solutions ( u ( λ ) , λ )) to F ( u ( λ ) , λ ) = 0 for λ ∈ V . Moreover, u ( λ ) is continuous with respect to λ in V . The IFT effectively states that if the linearization F u of the nonlinear operator operator F is nonsingular at the point [ u , λ ] , then there is a unique solution u ( λ ) for each λ in aball around λ . More details about this theorem and its proof can be found in [36, 51, 10].If F u is singular, however, the proof of the IFT fails, suggesting the possibility of twoor more u ( λ ) branches, or no solutions, for some λ in every neighborhood around λ .The form of the branching depends on the structure of the subspaces associated with thelinear maps F u ( u , λ ) and F λ ( u , λ ) . In the case of a “fold”, there is a one-dimensionalpath through [ u , λ ] ; in the case of a simple (or more general) singular point, there isthe possibility of branch-switching, with two (or more) branches of solutions crossingthrough [ u , λ ] .One of the central tools in analytic bifurcation theory is Lyapunov-Schmidt Reduc-tion [52]. To explain, assume F u ( u , λ ) is a Fredholm operator of index k , and thatdim ( N ( F u ( u , λ ))) = n . Define now projection operators P : X → X and Q : X → X with P ( X ) = N ( F u ( u , λ )) and ( I − Q )( Y ) = R ( F u ( u , λ )) . Equation (2.1) isequivalent to the pair ( I − Q ) F ( y + z, λ ) = 0 , (2.2) QF ( y + z, λ ) = 0 , (2.3)with y = ( I − P ) u and z = P u . Equation (2.2) satisfies the assumptions of the IFT, andso one obtains a unique solution branch y ( z, λ ) , then substitutes into (2.3) to obtain the branching equation : QF ( y ( z, λ ) + z, λ ) = 0 . (2.4)One then solves for z ( λ ) to get the branch u = y ( z ( λ ) , λ ) + z ( λ ) . In practice, one solves(2.2)–(2.3) by expanding the operators in bases of N ( F u ( u , λ )) and N ( F u ( u , λ ) ∗ ) .A more detailed description of the application of Lyapunov-Schmidt Reduction to varia-tions of the conformal method may be found in [26, 11]. One of our goals here is to applythe reduction technique to the far-from-CMC parameterizations that were described ear-lier. More information on this decomposition can be found in [36].2.2. Numerical Bifurcation Analysis.
To apply Lyapunov-Schmidt reduction analyti-cally, one needs detailed information about the null and range spaces of the linearizationoperators F u and F ∗ u , and therefore the technique is usually limited to model situations.However, by discretizing problem (2.1), it becomes tractable to explicitly compute theinformation one needs for a finite-dimensional approximation of (2.1). The problem re-tains the structure of (2.1), but the discretized problem now involves finite-dimensionalspaces X = Y = R n and Z = R m , where n is the resolution of the discretization(e.g. number of finite element basis functions), and m is the number of parameters. Onenow numerically computes bases explicitly for the range and null spaces of what arenow matrix operators F u and F λ . Moreover, a numerical continuation algorithm can be esigned around a predictor-corrector strategy: one increments the parameter λ → λ as part of a prediction step, followed by the use of Newton’s method to solve (2.1) tocorrect the solution u ( λ ) → u ( λ ) . Where the fold or higher-order singularity on thebranch is encountered, the linearization F u becomes singular, leading to failure of New-ton’s method at the correction step. To remedy this, one adds a normalization equation N ( u ( λ ( s )) , λ ( s ) , s ) = 0 that allows the larger coupled system involving F and N toagain be solvable. One of the standard normalization techniques is known as pseudo-arclength continuation , based on parameterizing λ ( s ) by arclength s . These numericaltechniques are well-studied [35] and there are well-established software packages thatimplement these techniques, such as AUTO [6]. We use AUTO as our primary tool inour numerical analysis of the conformal method for far-from-CMC seed data.3. N UMERICAL R ESULTS
The AUTO software package applies numerical bifurcation analysis to systems of or-dinary differential equations. Thus, to apply it to the conformal method, we require seeddata with sufficient symmetry so that the conformally-parameterized constraint equations(1.4)-(1.5) reduce to ODEs. Once this is done, the parameter space can be explored viahomotopies starting with CMC solutions. The following subsections describe a numberof concrete datasets where we have done so and report on folds and the number of solu-tions found as the mean curvature is made increasingly far-from-CMC and as the size ofthe TT tensor is varied. Some care is needed in interpreting our results. If we find, e.g.,two solutions for a given seed data set, it does not imply that there are only two. Rather,with homotopies we chose starting from CMC data, we were only able to find two. Theremay be more that we did not find because we did not, or were not able, to explore theparameter space more broadly. A similar caveat applies when we find no solutions; theremay be solutions that we did not find along our homotopies. Additionally, because weseek solutions having the same symmmetry as the seed data (in practice, these are solu-tions depending on only one coordinate of the underlying 3-manifold), we cannot ruleout the possibility of additional solutions that break this symmetry. Finally, because ofthe high symmetry of our seed data, our metrics always admit conformal Killing fields,and hence violate a key technical hypothesis of most theorems concering the conformalmethod with non-CMC seed data.3.1.
Sign-changing mean curvature.
In this section we examine properties of the con-formal method when the far-from-CMC regime is reached via a sign changing meancurvature. The conformal seed data has the following form: • The manifold is S × M where M is one of S , T or a compact quotient H of hyperbolic space. We use s for the unit speed parameter along S . • The mean curvature is τ = 1 + a cos( s ) . So a = 0 is the CMC case, and τ is sign-changing whenever | a | > . • We work with two different classes of TT tensors.(1) On a product ( M n , g ) × ( M n , g ) , the tensor ¯ σ = n g − n g (3.1)is easily seen to be transverse-traceless. For many of our experiments weuse a TT tensor of the form µσ , where µ is a constant.
2) Additionally, on S × T and on S × S we can find a TT tensor ˆ σ with con-stant (nonzero) norm that is pointwise orthogonal to L W for W = w ( s ) ds .These are easy enough to find on T and Appendix A describes a suitableconstruction on S . Some of our experiments make use of TT tensors of theform σ = η ˆ σ where η is a constant. • For simplicity of exposition, we use a lapse density N = 1 / . We conductedexperiments with other choices for the lapse density but did not see qualitativelydifferent phenomena.For solutions φ = φ ( s ) and W = w ( s ) ∂ s of the conformally-parameterized constraintequations (1.4)-(1.5) having the same symmetry as our seed data, the constraint equationsreduce to the coupled ODEs − ϕ (cid:48)(cid:48) + Rϕ + 23 τ ϕ = 23 ( µ + 2 w (cid:48) ) ϕ − + η ϕ − , (3.2) w (cid:48)(cid:48) = τ (cid:48) ϕ , (3.3)where R is the constant scalar curvature of the product manifold, and where we set η = 0 if R < .We start by examining solutions on S × T , and it turns out that the effects of the twoTT tensors ˆ σ (corresponding to the parameter η ) and ¯ σ (corresponding to the parameter µ ) are quite different. In particular, if η = 0 and hence σ = µ ¯ σ , it is easy to see that φ ≡ | µ | / and w = µa sin( s ) solve system (3.2). These exact solutions are among thosediscussed in [44], and are the only solutions we were able to find using AUTO. Hencewe obtain results consistent with existence and uniqueness for this family, save for theexceptional case µ = 0 where the solution degenerates to zero volume.On the other hand, fixing µ = 0 , Figure 1 indicates the multiplicity of solutions foundon S × T with τ = 1 + a cos( s ) and σ = η ˆ σ as the parameters a and η are varied. Thiscomputation is an analogue of the examples of [42] recalled in Theorem 1.3, except thatit involves a family of smooth, rather than piecewise constant, mean curvatures.In the region where the mean curvature changes sign, (i.e., for a > ) we find a fold,indicated by a solid blue line, and no solutions when the TT tensor is sufficiently large.Figure 2 indicates how the volume of the solution metric changes changes as we traversethe gray dashed lines of Figure 1; the plots, in green, indicate (cid:82) S φ , which agrees withvolume up to an inessential constant factor depending on the second factor of the productmanifold. On the vertical gray dashed line, corresponding to Figure 2 (left-hand side),the sign-changing mean curvature is fixed and the size of the TT tensor is varied. Whenthe TT tensor is sufficiently large a fold appears and there are no solutions, and as itis decreased to zero there are two solutions, one heading to zero volume and the otherblowing up. The horizontal gray dashed line of Figure 1 corresponds with Figure 2 (right-hand side) and we again observe the fold and a branch where the volume blows up. Itis difficult to ascertain from this graph the precise value of a where the blowup occurs,and computationally we found it difficult to approach the singularity. We find, however,that near the singularity the value of (cid:82) φ along this line is in reasonable agreementwith a growth rate ∼ ( a − − . . In later related examples we find more conclusiveevidence of blowup at a = 1 , so we infer this is the case here as well. That is, there isa transition at a = 1 , the threshold of sign-changing mean curvatures. The red dashedlines of Figure 1 indicate locations where we have inferred blowup occurs. The resultswe observe here are completely parallel with the prior analytical results of [42] found fora more restrictive mean curvature. .0 0.5 1.0 1.5 2.0 2.5 3.0 a η F IGURE
1. Multiplicity of solutions found on S × T . Seed data: τ =1 + a cos( s ) and σ = η ˆ σ . The blue line is a computed fold, whereas thered dashed lines indicate locations where blowup is inferred. The bluedotted line indicates a zero-volume solution, which should be discounted.Solutions along the gray dashed lines are discussed in Figure 2. η Z φ a F IGURE
2. Volume of solutions on S × T as the size η of the TT tensor(left-hand side) and as the mean curvature τ = 1 + a cos( s ) (right-handside) are varied. The left-hand graph corresponds with the vertical graydashed line of Figure 1 at a = 1 . , and right-hand graph correspondswith the horizontal gray dashed line at η = 1 . .5 1.0 1.5 2.0 2.5 a µ F IGURE
3. Multiplicity of solutions found on an S × S with R =0 . . Here τ = 1 + a cos( s ) and σ = η ˆ σ . The blue line is a computedfold. On the blue dotted line at µ = 0 (i.e., σ ≡ ) there is no solution;the volume has shrunk to zero. Solutions along the gray dashed lines arediscussed in Figure 4. η Z φ a F IGURE
4. Volume of solutions on an S × S with R = 0 . as the sizeof the TT tensor and as the mean curvature are varied. The left and right-hand graphs correspond with the vertical and horizontal gray dashed linesof Figure 3 respectively. The dashed lines are the corresponding volumeson S × T from Figure 2 for comparison. .5 1.0 1.5 2.0 2.5 a µ F IGURE
5. Multiplicity of solutions found on an S × H with negativescalar curvature R = − . . Seed data: τ = 1 + a cos( s ) and σ = µσ .The blue line is a computed fold, whereas the red dashed line indicatelocations where blowup is inferred. Solutions along the gray dashed linesare illustrated in Figure 6. µ Z φ a F IGURE
6. Volume of solutions on an S × M with R = − . as thesize of the TT tensor and as the mean curvature are varied. The left andright-hand graphs correspond with the vertical and horizontal gray dashedlines of Figure 5 respectively. ecalling that the only analytical results for sign-changing mean curvatures are avail-able in the Yamabe-null case, we now consider the effect of changing the Yamabe classin these computations. We first consider the Yamabe-positive case by adjusting the previ-ous computation by setting R = 0 . in system (3.2) (e.g. by working on an appropriate S × S ). As in the Yamabe-null setting, when working with families of seed data with σ = µ ¯ σ we found tame behavior (one solution was found for each parameter). On theother hand, for seed data with σ = µ ˆ σ and τ = 1 + a cos( s ) the situation is more com-plicated. Figure 3 indicates the multiplicity of solutions found for this family and canbe compared directly with its Yamabe-null counterpart, Figure 1. The region of zerosolutions has vanished and we find solutions exist always. However, in a region near theoriginal Yamabe-null fold, we find two folds and a narrow region of multiple solutions inbetween. Figure 4 shows the effect of traversing along the dashed gray lines of Figure 3and indicates how the family of solutions in this case corresponds with the Yamabe nullfamilies in Figure 2. Note that the various blowup phenomena found in the Yamabe-nullcase have vanished. The observed fold can be thought of as a purturbation of the situa-tion at R = 0 , and separate computations show that as R is pulled away from zero andapproaches, e.g., R = 1 the volume curves in Figure 4 stablize further, and the doublingback behavior vanishes.Turning to the Yamabe-negative case we set R = − . in equations (3.2) and use η = 0 since we do not have an equivalent for ˆ σ for Yamabe-negative seed data. Thus weuse µ to scale the size of the TT tensor, and unlike the Yamabe-positive and -null caseswhen using σ = µσ , we find interesting results; Figure 5 shows the number of solutionsfound when using mean curvatures of the form τ = 1 + a cos( s ) . Note that, unlike theparameter η , system (3.2) does not have even symmetry with respect to µ and henceour computations involved values of µ with both signs. As the mean curvature is madeincreasingly far-from CMC we find a fold, and subsequently no solutions. Just as in theYamabe-null case, the second branch of solutions blows up at a = 1 , the value of a thattransitions from constant-sign to sign-changing mean curvatures; (Figure 6, right-handside). On the other hand, far enough into the far-from-CMC regime we were unable tofind solutions of system 3.2. This is perhaps surprising since in the near-CMC settingone can always find solutions when σ ≡ (unless τ ≡ as well), and indeed solutionsat σ ≡ are a hallmark of Yamabe-negative CMC seed data. Instead, we find that at σ ≡ , as a is increased to make the solution far-from-CMC, there is a fold around a = 2 and no solutions were encountered beyond this point. The absence of solutions appearsto be loosely associated with the behavior when µ = 0 (i.e. σ ≡ ), although one notesthat the tip of the ‘nose’ on the blue fold line of Figure 5 does not lie on the line µ = 0 .Therefore, there are values of a where no solutions exist at µ ≡ , but for which solutionsexist for certain values of µ (cid:54) = 0 .3.2. Constant-sign mean curvatures.
We now examine excursions into the far-fromCMC regime using mean curvatures of the form τ = ξ a , where ξ is a positive function.Starting again with S -dependent data of the form of the previous section, but now withmean curvature τ ( s ) = ( − cos( s )) a , we were only able to find a single solution ofthe constraint equations for all choices of a , µ and η , except (as is expected) when σ ≡ in the Yamabe non-negative case. We can understand the tame behavior we observedby appealing to the limit equation of Theorem 1.2. For S -symmetric solutions of the S -dependent data we consider, the limit equation becomes (cid:18) W (cid:48) N (cid:19) (cid:48) = α (cid:12)(cid:12)(cid:12)(cid:12) W (cid:48) N (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:48) τ (3.4) nd it is straightforward to show that this admits no solutions on S . Theorem 3.1.
Let τ > be in C . Then there are no nontrivial C solutions W of the S -dependent limit equation (3.4) .Proof. Suppose W is a nontrivial solution, and consider a maximal interval on which W (cid:48) does not vanish. On this interval we have log( W (cid:48) ) (cid:48) = k log( τ ) (cid:48) (3.5)where k = α if W (cid:48) > on the interval, and k = − α if W (cid:48) < . Hence W (cid:48) = cτ k (3.6)for some constant c (cid:54) = 0 . At the endpoints of the interval W (cid:48) tends to zero. But theright-hand side of equation (3.6) is uniformly bounded away from zero. (cid:3) Strictly speaking, Theorem 1.2 does not apply in our setting because of the presenceof conformal Killing fields. We expect, however, that one can use techniques found in[42] to adapt the main theorem of [17] to this specific family of data to conclude that thenonexistence of solutions to (3.4) implies existence of ( S -symmetric) solutions.The simple behavior seen for S -dependent data does not hold generally, however.Throughout the remainder of this section we use conformal seed data of the followingtype: • The manifold is S × M where M is one of S ,or H . We use φ ∈ (0 , π ) for alatitude parameter on S . • The mean curvature τ = ξ a depends only on φ , where ≤ ξ ≤ and ξ ( φ ) = 1 somewhere. In practice we used ξ ( φ ) = 23 −
13 cos( kφ ) with k = 1 or . • The TT tensor is σ = µσ , where σ was introduced in equation (3.1). • The lapse density is N = 1 / .As a first example, consider data on S × S . The reduced conformal constraint equa-tions for latitude-dependent data are more complicated than those for S -dependent data;the main differential operators are ∆ f = 1 r ( f (cid:48)(cid:48) + cot φf (cid:48) ) , and div L W = (3 w (cid:48)(cid:48) + 3 cot φw (cid:48) + 12 r (1 − φ ) w ) dφ where f = f ( φ ) and W = w ( φ ) dφ . We seek solutions of the constrant equations (1.2)-(1.3) the form ϕ = ϕ ( φ ) and W = w ( φ ) dφ supplemented with boundary conditions ϕ (cid:48) =0 and w (cid:48) = 0 at φ = 0 and φ = π needed to ensure regularity. An additional boundarycondition w = 0 at φ = 0 , π is effectively enforced by the momentum constraint.Figure 7 shows the number of solutions found on an S × S with scalar curvature R = 1 with σ = µσ and a relatively simple non-CMC mean curvature τ = (cid:20)
23 + 12 cos( φ ) (cid:21) a Theorem 1.5 would apply to this data, except for the usual caveat about conformal Killingfields and, more crucially, the fact that the mean curvature violates the non-generic in-equality (1.7). We nevertheless find behaviour consistent with its conclusions: multiple a µ a F IGURE
7. Multiplicity of solutions found on an S × S with positivescalar curvature R = 1 . Seed data: τ = ( + cos( φ )) a and σ = µσ . Thesolid blue line is a computed fold, whereas the red dashed line indicatelocations where blowup is inferred. At the dotted blue line at µ = 0 thereis a zero volume solution which should be discounted. The gray dashedlines are discussed in Figure 8.
100 50 0 50 100 µ Z φ a F IGURE
8. Volume of solutions on an S × S with R = 1 as the size µ of the TT tensor and as the mean curvature ξ a is varied. The left- andright-hand graphs correspond with the vertical and horizontal gray dashedlines of Figure 7 respectively. π/ π/ π/ πφ F IGURE
9. Ratio of (cid:113) ϕ τ to (cid:12)(cid:12) N L W (cid:12)(cid:12) as a function of latitude φ forthe large solution of the conformally-parameterized constraint equationsat a = 2 . , µ = 1 in Figure 7. The ratio is nearly 1, indicating that thevector field W of the solution is nearly a solution of the limit equation(1.6).solutions when both a is sufficiently large and µ (cid:54) = 0 is sufficiently small. Moreover, thetransition to the far-from-CMC regime is abrupt, starting at a ≈ . , which we will call a ∗ . Figure 8, (right-hand side) shows that a ∗ is associated with a blowup of a branch ofsolutions, and one expects there is a solution of the limit equation (1.6) with α = 1 at a = a ∗ . Indeed, the limit equation arises as the Hamiltonian constraint degenerates tothe algebraic equation (cid:114) φ τ = (cid:12)(cid:12)(cid:12)(cid:12) N L W (cid:12)(cid:12)(cid:12)(cid:12) , (3.7)which can then be substituted back into the momentum constraint. Figure 9 shows theratio of the two sides of equation (3.7) for the larger of the two solutions at the point a = 2 . m µ = 1 in Figure 7. The ratio is nearly 1, so the solution vector field W at thatpoint is nearly a solution of the limit equation.The proofs of Theorems 1.4 and 1.5 demonstrate the existence of multiple solutionsfor a large and µ small by showing that at µ = 0 there is both a zero volume solution anda true solution, and that perturbing off of these yields two solutions; this mechanism isillustrated in Figure 8 (left-hand side). Recall that solutions with µ = 0 are impossible inthe near-CMC setting for Yamabe-positive seed data such as this, and at the dotted lineat µ = 0 of Figure 7 there is one less solution than at neighboring points (so there is nosolution to the left of the singularity at a = a ∗ and just one solution to the right).As a test of the robustness of these results, consider the same conformal seed data asthe previous example, except that the mean curvature is now τ = (cid:20)
23 + 12 cos(2 φ ) (cid:21) a . a µ a F IGURE
10. Multiplicity of solutions found on an S × S with positivescalar curvature R = 1 . Seed data: τ = ( + cos(2 φ )) a and σ = µσ ;note the in the argument of cos . The solid blue line is a computed fold,whereas the red dashed lines indicate locations where blowup is inferred.At the dotted blue line there is a zero-volume solution, which should bediscounted. The gray dashed lines are discussed in Figure 11. µ -1 Z φ a -1 F IGURE
11. Volume of solutions on an S × S with R = 1 as the size µ of the TT tensor (left-hand side) and as the mean curvature ξ a (right-hand side) are varied. The left- and right-hand graphs correspond with thevertical and horizontal gray dashed lines of Figure 10 respectively. igure 10 illustrates the multiplicity of solutions we found as we varied the parameters µ and a . Here again we find a sharp transition to the far-from-CMC setting, now at a = a ∗ ≈ . . There is blowup associated with this transition (Figure 11, right-hand side),and again we presume there is a solution to the limit equation with α = 1 and a = a ∗ .For a > a ∗ we find a solution at µ = 0 in addition to the zero volume solution (Figure 11,left-hand side), and hence there two solutions for µ sufficiently small. However, we findan apparent difference between scaling µ large and positive versus large and negative forthis seed data. For µ positive and large enough (depending on a ) there are no solutions,just as was the case in Figure 7. But for µ large and negative we were unable to find afold and a consequential transition to zero solutions. Instead, the two solutions remainwell-separated in volume as µ is made large (Figure 11, left-hand side). We cannotrule out the possibility that the two branches eventually merge, but this was not thecase out to µ = − . Although this data violates inequality 1.7 of Theorem 1.5,our observations are nevertheless consistent with its conclusions. In particular, unlikeTheorem 1.4, Theorem 1.5 does not predict non-existence for large TT tensors, and it isconceivable that 1.5 holds more generally for mean curvatures not satisfying inequality1.7.The previous two examples involve Yamabe-positive data. To explore the other Yam-abe classes we consider latitude-dependent data on S × H ; by varying the size of theround S factor we can obtain any desired constant scalar curvature. Before looking atthe other Yamabe classes, however, we remark that even for Yamabe-positive data of thistype we find differences from what was observed for S × S . Figure 12 illustrates themultiplicity of solutions found on an S × H with R = 0 . , where the seed data has σ = µσ and τ = (cid:20) −
13 cos( φ ) (cid:21) a . This seed data is comparable to that used for Figure 7, but there are some fine differencesin the results. Again we find a sharp transition to the far-from-CMC setting, now at a = a ∗ ≈ . . However, for a > a ∗ the multiplicity of solutions is a bit complicated.There is a primary fold roughly at a = 3 . in the far-from-CMC regime. Beyond thispoint, the behavior is similar to that of Figure 7, with no solutions when µ is sufficientlylarge and two solutions when µ (cid:54) = 0 is sufficiently small. Between a = a ∗ and a = 3 . ,however, the situation has changed from that of Figure 7. We found variously between1 and 4 solutions, and no convincing evidence that there are no solutions when µ issufficiently large at, e.g., a = 2 . .Figure 13, illustrates how the the various folds appear as the dashed line at µ = 7 inFigure 12 is traversed. First the right-most purple fold is encountered, then the left-mostand finally the blue fold around a = 3 . is hit before heading off to the singularity at a = a ∗ . Conversely, Figure 14 illustrates the various solutions along the line a = 3 . , andwe see two disconnected loops (the upper branches in Figure 14, left-hand side, rejoinat µ ≈ − ). As was discussed previously for Figure 8, Figure 14 illustrates howone can visualize the multiple solutions near µ = 0 as perturbations of the usual zerosolution at µ = 0 and the additional three non-trivial solutions we found there.Turning to non-Yamabe-positive seed data, Figure 15 illustrates the number of solu-tions found with the same conformal data as in Figure 12, except now with R = − . and R = 0 . For this seed data there are no applicable far-from-CMC theorems to guideexpectations, and we find a situation similar to the Yamabe-negative sign-changing dataof Figure 5. There is a sharp transition to far-from-CMC data at a ∗ ≈ . , but nowsolutions vanish sufficiently far into the far-from CMC regime. The blue fold from the .5 3.0 3.5 4.0 4.5 5.0 a µ F IGURE
12. Multiplicity of solutions found on an S × H with positivescalar curvature R = 0 . . Seed data: τ = ( + cos( φ )) a and σ = µσ . The solid blue and purple lines are computed folds, whereas thered dashed line indicate locations where blowup is inferred. At the bluedotted line at µ = 0 one solution has zero volume and should be ignored.Solutions along the gray dashed lines are discussed in Figures 13 and 14. a -1 Z φ F IGURE
13. Volume of solutions on an S × H with R = 0 . as themean curvature ξ a (right-hand side) is varied. Solutions correspond withhorizontal gray dashed line at µ = 7 in Figure 12. At a = 3 . , foursolutions are found for a single conformal seed dataset. µ -4 -3 -2 -1 Z φ µ F IGURE
14. Left-hand side: volume of solutions on an S × H with R = 0 . as the size µ of the TT tensor is varied. Solutions correspondwith the vertical gray dashed line at a = 3 . in Figure 12. Right-handside shows a detail of the small loop near µ = 0 . The large upper loopeventually closes at µ ≈ − . a µ a F IGURE
15. Multiplicity of solutions found on an S × H with scalarcurvatures R = − . (left-hand side) and R = 0 (right-hand side). Seeddata: τ = ( − cos( φ )) a and σ = µσ . The solid blue lines are computedfolds, whereas the red dashed lines indicate locations where blowup isinferred. The folds are comparable with the blue fold of Figure 12, whichhas a similar shape when drawn at this scale. On the right-hand side( R = 0 ), the dotted blue line indicates a zero solution which should bediscounted. Fine details near µ = 0 in the region . < a < . arepotentially unresolved; see Figure 16 (middle). .0 0.2 0.4 0.6 0.8 1.0 a Z φ -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 F IGURE
16. Volume of solutions along σ ≡ for an S × H having R = − . (left), R = 0 (middle) and R = 0 . (right). Mean curvature τ = ( − cos( φ )) a . Conformal seed data is the same as in Figures 12and 15.Yamabe-positive data of Figure 12 persists, but the purple fold that extended out alongthe line µ = 0 has vanished. The apparent non-existence of solutions far enough into thefar-from-CMC regime violates the conclusion of Theorem 1.5, and we therefore suspectthat the Yamabe-positive hypothesis of Theorem 1.5 is essential.Although the vertical scales for Figures 12 and 15 are markedly different, it is not thecase that we have missed a fine feature near the line µ = 0 in Figure 15 corresponding tothe purple folds of Figure 12. Indeed, Figure 16 illustrates the volume of solutions alongthe line µ = 0 (i.e. σ ≡ ) for the three cases R = − . , R = 0 and R = 0 . illustratedin Figures 12 and 15. For far-from-CMC Yamabe-positive seed data (Figure 16, right-hand side) we find solutions at σ ≡ ; these solutions are not present for near-CMC dataand had not been expected before [45]. By contrast, for the Yamabe-negative seed datathe near-CMC solutions at σ ≡ vanish once the mean curvature is sufficiently far-fromCMC (Figure 16, left-hand side). At the boundary R ≡ (Figure 16, middle) we finda narrow band in which solutions exist; none in the near-CMC case (which is expected)and none for the far-from CMC seed data as well. The narrow band is reminiscent of thewell known one-parameter family of solutions found found for CMC Yamabe-null seeddata when τ ≡ and when σ ≡ . It is also similar to the somewhat analogous one-parameter families found for particular Yamabe-null seed data in [42] and [44]. Whatwould have been a vertical line of solutions in the analytic cases has been deformed tothe distorted vertical line of Figure 16, middle.The role of solutions at σ ≡ appears to be important for understanding the conformalmethod in the far-from-CMC setting, and we believe this is a consequence of σ beingabsent from the limit equation. Nevertheless, the interplay between σ ≡ and σ (cid:54)≡ is nuanced. For example, the breakdown of the existence of solutions along σ ≡ forYamabe-negative data (Figure 16, left-hand side) at a ≈ . is not perfectly correlatedwith nonexistence of solutions for nearby values of a . This can be seen by inspectingthe blue fold of Figure 15, left-hand side. It crosses µ = 0 at a value a ≈ . but therenevertheless exist certain solutions for values of a > . beyond the crossing point, butnot much larger.Finally, we consider seed data on S × H with σ = µσ and τ = (cid:20) −
13 cos(2 φ ) (cid:21) a , a µ a F IGURE
17. Multiplicity of solutions found on an S × H with scalarcurvatures R = 0 (left-hand side) and R = 0 . (right-hand side). Seeddata: τ = ( − cos(2 φ )) a and σ = µσ . The solid blue line indicatesa fold, whereas the red dashed line indicate locations where blowup isinferred. The blue dotted lines at µ = 0 indicate one solution has zerovolume and should be ignored.which can be compared with the seed data used for Figure 10. Figure 17 illustrates thenumber of solutions found for R = 0 and R = 0 . , and the outcomes are qualitativelysimilar to those of Figure 10. In particular, for µ < we do not find evidence of non-existence of solutions. On the other hand, for this same data but R = − / we obtainmultiplicities shown in Figure 18 and find a situation akin to what we have seen previ-ously in Figure 5 and Figure 15 (left-hand side) for Yamabe-negative data: no solutionsfor a sufficiently large. In an independent computation, not shown, we found that thelocation of the blue fold crossing µ = 0 (e.g. at a ≈ in Figure 18) grows as ( − R ) − and therefore we believe there is indeed a transition at R = 0 between the two qualitativebehaviors seen here for R = 1 / and R = − / .4. D ISCUSSION
The limit equation (1.6) appears to play a central role in the solution theory of theconformal method, at least for mean curvatures that do not change sign. In the caseswhere we could show that there is no solution of the limit equation with the symmetryof the data ( S -dependent seed data on S × M ), we were also unable to find behaviorthat was any different from the near-CMC theory. For the remaining cases where weinvestigated constant-sign mean curvatures τ = ξ a , there was a singular value a ∗ . For a > a ∗ we found differences from the near-CMC theory: multiple solutions or non-existence of solutions were the general rule, with exceptions occuring only at transitionssuch as folds. As a → a ∗ from above, we found solutions of the constraint equationswith volumes that appeared to approach infinity; in particular, it was always possible tofind such solutions with σ ≡ We conjecture that at a = a ∗ there is a solution of thelimit equation with α = 1 , in which case for each a > a ∗ there is also a solution of thelimit equation with α = a/a ∗ < . That is, far-from-CMC behavior appears to occurprecisely when a solution to the limit equation exists for some α ∈ (0 , .For sign changing mean curvatures, we found a corresponding transition to far-from-CMC phenomena once the mean curvature changed sign. This was certainly true forthe Yamabe-negative and Yamabe-null data we examined, and at least weakly so for theYamabe-positive data of Figure 3 where narrow folds occurred, but unique solutions were
20 40 60 80 100 120 140 a µ F IGURE
18. Multiplicity of solutions found on an S × H with scalarcurvatures R = − . Seed data: τ = ( − cos(2 φ )) a and σ = µσ . Thesolid blue indicates a fold, whereas the red dashed line indicate locationswhere blowup is inferred.more typical. In any event, in these examples, non-standard behaviour only occurred forconformal seed data where the mean curvature changed sign. In may be a that a betterparameterization for sign-changing mean curvatures, different from τ = 1 + aξ , wouldprovide a sharper, more definitive transition.Although Theorem 1.1 does not apply, strictly speaking, to our examples (as theypossess nontrivial conformal Killing fields), our observations were consistent with it.Whenever we worked with Yamabe-positive seed data we were able to find solutions ofthe constraint equations, so long as σ (cid:54)≡ was small enough: Figures 3, 7, 10, 12 and17 (right-hand side). Conversely, except for cases where we could show that there wasnot a solution of the limit equation, far-from-CMC Yamabe-negative seed data lead tonon-existence sufficiently far into the non-CMC zone: Figures 6, 15 (left-hand side) and18. This was true even at σ ≡ , in stark contrast to the near-CMC theory.The situation for Yamabe-null data is harder to characterize. Sometimes it behavedlike Yamabe-positive data (Figures 1 and 17, left-hand side), with solutions existing forsufficiently small TT tensors. Sometimes it behaved like Yamabe negative data (Figure15, right hand side), with solutions vanishing far enough into the far-from CMC zone.Moreover, the analytical work of [42] shows that other variations are also possible.The conclusions of Theorem 1.5 were found to hold generally, even for mean curva-tures that violate inequality 1.7, so long as there appeared to be a solution of the limitequation. That is, for far-from-CMC Yamabe positive data (with constant-sign mean cur-vature), we found that there were at least two solutions when the TT tensor was smallenough (and that there was at least one corresponding nonzero solution at σ ≡ ).On the other hand, Theorem 1.4, which also describes non-existence for Yamabe-positive far-from-CMC seed data when the TT tensor is large, was not found to holdin general. Indeed, we found a hodge-podge of apparent non-existence phenomena on amabe-positive seed data. Sometimes there was an immediate onset of nonexistencebehavior in the far-from-CMC zone (Figure 7). Sometimes nonexistence was brought onby scaling the TT by a large constant of one sign, but not for large constants of the othersign (Figures 10 and 17 (right-hand side)). In one case (Figure 12) there appeared to becertain far-from-CMC seed data that did not lead to nonexistence regardless of how largethe TT tensor was scaled. And in the sign-changing Yamabe-positive case we examined,existence appeared to be pervasive (Figure 3). It is similarly hard to pin down precisenon-existence behavior for the Yamabe-null seed data we examined.We saw no apparent rule to describe the profiles of the various folds we saw. Thatis, we were unable to discern anything that might help concretely predict the thresholdof non-existence when scaling the TT tensor or the specific number of solutions corre-sponding to given seed data; although zero, one or two solutions were typical, sometimesthere were more. 5. C ONCLUSION
Our numerical work suggests that the conformal method appears to suffer from per-vasive drawbacks as a parameterization of vacuum, non-CMC solutions of the constraintequations. At least among the data we considered, the general rule was multiple solutionsor no solutions at all once the conformal seed data was sufficiently far-from-CMC. Be-cause of the limitations of AUTO, we were only able to examine highly symmetric seeddata, and we therefore only probed a select few, very special examples. Nevertheless, itis difficult to imagine that the many cases of multiple solutions we found are not stableunder small perturbations of the metric that violate symmetry.Our results suggest a couple of theorems that might be reasonable targets for fu-ture efforts. For example, can non-existence of solutions for sufficiently far-from-CMCYamabe-negative seed data be established? However, we caution that it is possible thatdescribing the details of the conformal method for far-from-CMC data will lead to afuller understanding of the conformal method, but also to nothing useful about generalrelativity. Unless there is some physics associated with to the multiplicity of solutions orthe various shapes of the folds, it may simply be that the conformal method is an excel-lent parameterization of the CMC solutions of the constraints that breaks down as a charton the larger constraint manifold. Since the conformal method is the only general toolavailable for constructing solutions of constraint equations de novo , it raises the ques-tion of whether a suitable alternative parameterization for non-CMC initial data exists.One potential was proposed in [38] and examined for near-CMC constructions in [25],but its properties in the far-from-CMC setting are unknown and the broader question offinding a well-behaved global parameterization of solutions of the constraint equationsis essentally open. 6. A
CKNOWLEDGMENTS
JD was supported in part by NSF DMS/RTG Award 1345013 and NSF DMS/FRGAward 1262982. MH was supported in part by NSF DMS/FRG Award 1262982 andNSF DMS/CM Award 1620366. TK was supported in part by NSF DMS/RTG Award1345013. DM was supported in part by NSF DMS/FRG Award 1263544. PPENDIX
A. A
CONSTANT NORM TT TENSOR ON S × S In this section we construct a transverse-traceless tensor on S × S that has con-stant norm and is pointwise orthogonal to L W when W is an S -dependent vector fieldpointing along S .Consider normal (polar) coordinates ( r, θ ) on the unit round sphere S centered at thenorth pole, so that the metric has the form g = dr + sin ( r ) dθ . Let ω = sin( r ) dθ . Itis clear that ω fails to be continuous at the north and south poles of S , but is otherwisesmooth. A straightforward computation shows that |∇ ω | = cot( r ) . The singularity at r = 0 is O ( r − ) , with similar remarks holding at r = π . Hence ω ∈ W ,p ( S ) for any p < . Moreover, div ω = 0 in the region where ω is smooth (i.e. almost everywhere)and therefore ω is weakly divergence free.Now let s denote a unit speed parameter on S and set ˆ σ = ω ⊗ ds + ds ⊗ ω on S × S . Clearly σ is trace-free. Moreover since ω and ds are both divergence free, σ is a transverse-traceless tensor on S × S . Finally, | σ | = 2 | ω | | dz | + 4 (cid:104) ω, dz (cid:105) = 2 | ω | = 2 sin ( r ) | dθ | = 2 sin ( r )(sin( r )) − = 2 , except at r = 0 and r = π . That is, | σ | = 2 almost everywhere. Hence ˆ σ is a constant-norm W ,p transverse-traceless tensor on S × S for any p < . Although this level ofregularity is not ideal, it falls within the a category of regularity easily handled for theconformal method (e.g. [8]).If W = w ( s ) ∂ s then L W = 2 w (cid:48) ( ds ⊗ ds − g ◦ ), where g ◦ is the round metric on thesphere. Since ˆ σ only has ds ⊗ dθ and dθ ⊗ ds components, it is pointwise orthogonal toany such L W . R EFERENCES [1] P. T. Allen, A. Clausen, and J. Isenberg. Near-constant mean curvature solutions of the Einstein con-straint equations with non-negative Yamabe metrics.
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