ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Pages 52–63 (July 19, 2000) S 1079-6762(00)00081-0

ON THE CONNECTEDNESS OF THE SPACE OF INITIAL DATA FOR THE EINSTEIN EQUATIONS

BRIAN SMITH AND GILBERT WEINSTEIN

(Communicated by Richard Schoen)

Abstract. Is the space of initial data for the Einstein vacuum equations connected? As a partial answer to this question, we prove the following result: Let ℳ be the space of asymptotically flat metrics of non-negative scalar curvature on ℝ^{3} which admit a global foliation outside a point by 2-spheres of positive mean and Gauss curvatures. Then ℳ is connected.

Introduction

The Einstein vacuum equations of general relativity read:

R̄_{μν} − \frac{1}{2}R̄ḡ_{μν} = 0, (1)

where R̄_{μν} is the Ricci curvature tensor of a Lorentzian 4-manifold, and R̄ the scalar curvature. The basic problem for these equations is the Cauchy problem: given data on a time-slice M, consisting of a Riemannian metric g and a second fundamental form k on M, find the evolution of space-time according to (1). Not all the equations in (1) are evolution equations. Using the twice-contracted Gauss equation and the Codazzi equations of the Riemannian submanifold M, one finds that the normal-normal and normal-tangential components of (1) are:

(2)  R − |k|^{2} + (tr k)^{2} = 0,
(3)  ∇^{j}k_{ij} − ∇_{i} tr k = 0,

where R is the scalar curvature of M, and k its second fundamental form. These equations, called the Vacuum Constraint Equations, involve no time derivatives and hence are to be considered as restrictions on the data g and k; see [14]. We will only consider asymptotically flat (AF) solutions of these equations, i.e., solutions satisfying the decay:

g_{ij} − δ_{ij} = O(r^{−1}),
k_{ij} = O(r^{−2}),
R ∈ L^{1}(M).
\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

Received by the editors May 27, 1999.

2000 Mathematics Subject Classification. Primary 83C05; Secondary 58G11.

This research was supported in part by NSF grant DMS 9704760.

It is standard to choose the maximal gauge tr k = 0 in (2)–(3), which, as shown by Bartnik [1], involves no loss of generality. In this case, we get the Maximal Gauge Vacuum Constraint Equations:

(4)  R = |k|^{2} ,
(5)  tr k = 0,
(6)  div k = 0.

These form an underdetermined system of elliptic equations on M for g and k.

Much work has been devoted to finding solutions of (4)–(6); see for example [3, 4, 6, 7, 9, 10, 15] and the references therein. However, certain fundamental questions remain unanswered. For example, it is not known whether the space of AF initial data on a given 3-manifold M is connected, not even in the case M = ℝ^{3}. Since the evolution equations trace a continuous path in the phase space of initial data, either answer to this question would be of considerable significance for the dynamics of the Einstein equations.

The standard method for solving (4)–(6) has been the conformal method. In this method the free data is the conformal class of an asymptotically flat Riemannian metric g, and a trace-free divergence-free symmetric 2-tensor k on M. Since the trace-free and divergence-free conditions on k are invariant under the transformation g ↦ ϕ^{4}g, k ↦ ϕ^{−2}k, it suffices to find ϕ so that (4) is satisfied. This will be so provided that the Lichnerowicz equation is satisfied:

Δϕ − \frac{1}{8}Rϕ + |k|^{2} ϕ^{−7} = 0.

A solution of this equation can be found if the negative part of the scalar curvature is small enough in the L^{3/2} norm; see [9]. In particular, the question above can be reduced to the following purely geometric problem: is the space of AF metrics of non-negative scalar curvature on a 3-manifold M connected?

In this paper, we announce, and sketch the proof of a result which gives a partial answer in the affirmative to the question posed above; details will appear in [13]. We say that a topological 2-sphere S in M is quasiconvex if both the Gauss and the mean curvature of S are positive [8]. Let ℳ be the space of C_{−1}^{2,α} metrics g on ℝ^{3} with non-negative scalar curvature R ∈ L^{1} which admit a global coordinate system whose coordinate spheres are quasiconvex, and which satisfy in this coordinate system:

g_{ij} − δ_{ij} = O(r^{−1})  as r → ∞,
g_{ij} − δ_{ij} = 0  at r = 0.

The C_{−1}^{2,α} topology on ℳ is generated by the following system of neighborhoods of any metric g ∈ ℳ:

{g′ ∈ ℳ: ∑_{α=0}^{2} sup |(1 + r)^{m+1}∂^{m}(g_{ij} − g′_{ij})| + [∂^{2}g_{ij} − ∂^{2}g′_{ij}]_{α,−3} < ϵ} ,

where

[f]_{α,−k} = sup_{r} ((1 + r)^{k+α} sup_{x,y∈B_{r}} \frac{|f(x) − f(y)|}{|x − y|^{α}})

is the weighted Hölder norm with exponent α of f on ℝ^{3}. In fact, in view of the general covariance of the Einstein Equations, we are only interested in the quotient of ℳ by the group 𝒢 of diffeomorphisms of ℝ^{3}.

Main Theorem. The quotient of ℳ/𝒢 is path connected in the quotient topology induced by C_{−1}^{2,α} on ℳ.

Of course, this raises the following question: when does an AF metric g of nonnegative scalar curvature belong to ℳ? Clearly, a necessary condition is that g possesses no compact minimal surfaces. However we do not even know whether the absence of compact minimal surfaces suffices to guarantee the existence of a global foliation with positive mean curvature.

To prove our Main Theorem, we generalize a method introduced by Bartnik [2] to construct quasispherical metrics of prescribed scalar curvature. A metric is quasispherical if it can be foliated by round spheres, spheres of constant curvature. Bartnik observed that prescribing scalar curvature for this type of metric could be viewed as a parabolic equation on the sphere for one of the metric coefficients, u = |∇r|^{−1}, where r is the foliating function, provided that the mean curvature was also positive. We combine this with the Poincaré Uniformization as in [8] to get a general method to prescribe scalar curvature for metrics in ℳ. As an application, we prove the Main Theorem.

Denote by r the foliating function normalized so that the area of the spheres is 4πr^{2}, and by γ the induced metric on the spheres. Any smooth enough metric g ∈ ℳ can be written as:

g = u^{2}dr^{2} + e^{2v}γ̄_{AB}(β̂^{A}dr + rdθ^{A})(β̂^{B}dr + rdθ^{B}), (7)

where (θ^{1}, θ^{2}) are local coordinates on 𝕊^{2}, γ̄_{AB} is a fixed (independent of r) round metric of area 4π, and β̂ = β̂^{A}∂_{A} is the shift vector. Here, and throughout, we use the summation convention: repeated indices are summed over their range, 0, 1, 2, 3 for Greek indices, 1, 2, 3 for lower case Latin indices, and 1, 2 for upper case Latin indices. Let χ be the second fundamental form, H = tr_{γ} χ be the mean curvature of the spheres, and Π = ℒβ̂γ be the deformation tensor of β̂ on the spheres; then it can be checked that

(8)  χ̄ = ruχ = ((1 + rv_{r})γ − Π/2) ,
(9)  H̄ = ruH = (2 + 2rv_{r} − e^{−2v} divγ̄ β) ,

where β = e^{2v}β̂. It is important to note that both \left|{ \atop χ̄}\right|_{γ}^{2} and H̄ can be calculated in terms of only β, v, r, and the round metric γ̄ on 𝕊^{2}. Let N be the outer unit normal to the foliation spheres, let N̄ = ruN = r∂_{r} − β̂, let Δ /_{γ} be the Laplacian on the spheres with respect to γ, and let

κ = r^{−2}e^{−2v}(1 − Δ /v) (10)

be the Gauss curvature of the spheres. Then the equation for the scalar curvature R of g can be written as

H̄∂N̄u = r^{2}u^{2}Δ /_{γ}u + Āu − B̄u^{3}, (11) where

Ā = ∂N̄ H̄ − H̄ + \frac{1}{2} \left|{ \atop χ̄}\right|_{γ}^{2} + \frac{1}{2}H̄^{2},

B̄ = r^{2}(κ − \frac{1}{2}R) = e^{−2v}(1 − Δ /v) − \frac{1}{2}r^{2}R.

Noting that the Laplacian with respect to γ̄ is Δ / = r^{2}e^{2v}Δ /_{γ}, we obtain, provided that H > 0, the following Bernoulli-type parabolic equation for u on the unit sphere:

r∂_{r}u − β · ∇ /u = Γu^{2}Δ /u + Au − Bu^{3}, (12)

where ∇ /u is the tangential component of the gradient of u , Γ = e^{−2v}/H̄, A = Ā/H̄ and B = B̄/H̄. It follows from the comment following equations (8)–(9) that the coefficients Γ, A and B can be calculated in terms of only β, v, r, the round metric γ̄ on 𝕊^{2}, and R. The quasispherical case can be recovered by setting v = 0, and κ = 1, see [2].

The proof of the Main Theorem is based on the study of equation (12). The deformation to a flat metric is accomplished in several steps. First, the metric is smoothed out with the scalar curvature R truncated to be compactly supported. Next, we deform the metric to one satisfying 2κ > R. Then, we deform to a metric with compactly supported β and v. Finally, we deform to a flat metric. The last three steps are all based on the following strategy. The deformation g_{λ} is defined explicitly on a ball B_{r_{0}}. In the exterior of B_{r_{0}} we consider β_{λ}, v_{λ}, and R_{λ} as free data, and solve equation (12) on [r_{0}, ∞) × 𝕊^{2} for u_{λ} with initial conditions u_{λ}|_{S_{r_{0}}} . In order for this to be feasible, and for the resulting metric g_{λ} to yield a continuous path, we must ensure that β_{λ}, v_{λ}, and R_{λ} are continuous in the appropriate spaces, that H̄_{λ} is positive, and that R_{λ} is non-negative. In addition, one must verify conditions that guarantee the global existence of the solution u_{λ}, its appropriate decay as r → ∞, and continuity with respect to λ. The regularity of u_{λ} across S_{r_{0}} is obtained by solving equation (12) on [r′, r_{0} + ϵ) × 𝕊^{2}, 0 < r′ < r_{0}, with initial data u_{λ}|_{S_{r′}} and by using the uniqueness and regularity of solutions.

The plan of the paper is as follows. In the next section, we derive equation (11). In Section 2, we collect the analytical results we need on existence, uniqueness, asymptotic behavior, and continuous dependence on parameters of solutions of (12). Then in Section 3, we sketch the proof of the Main Theorem by deforming any metric in ℳ to the Euclidean metric.

1. The scalar curvature of a 3-manifold foliated by spheres

In this section, we derive equation (11). First, note that ∇r = u^{−1}N, and consequently u^{−1} = N^{i}∇_{i}r. It follows that

∇_{N}N = −u^{−1}∇ /u, (13)

where ∇ /u is the tangential part of ∇u. Furthermore, using N^{j}∇_{i}N_{j} = 0, we get

∇_{i}∇_{j}u^{−1} = N^{k}∇_{i}∇_{j}∇_{k}r + (∇_{j}∇_{k}r)(∇_{i}N^{k})
= N^{k}∇_{i}∇_{k}∇_{j}r + ∇_{j}(u^{−1}N_{k})(∇_{i}N^{k})
= N^{k}∇_{k}∇_{i}∇_{j}r + u^{−1}R_{jlik}N^{k}N^{l} + u^{−1}(∇_{j}N_{k})(∇_{i}N^{k}),

which after tracing with respect to g leads to

Δu^{−1} = ∇_{N}Δr + u^{−1} Ric(N, N) + u^{−1} |∇N|^{2} , (14)

where Ric is the Ricci tensor of g. We now need to calculate the first and third terms on the right hand side of (14). Since the tangential part of ∇N is the second fundamental form χ, and the normal part is given by (13), it is easy to see that:

|∇N|^{2} = |χ|_{γ}^{2} + u^{−2}|∇ /u|_{γ}^{2}.

Next, using equation (13) again, we find:

∇_{N}Δr = N^{k}N^{i}∇_{k}∇_{i}u^{−1} + (∇_{N}N^{i})(∇_{i}u^{−1}) + ∇_{N}(u^{−1}∇^{i}N_{i})
= ∇^{2}u^{−1}(N, N) + u^{−3}|∇ /u|_{γ}^{2} + ∇_{N}(u^{−1}H).

Substituting the last two equations into (14) gives

Δu^{−1} = ∇^{2}u^{−1}(N, N) + 2u^{−3}|∇ /u|_{γ}^{2} + u^{−1} Ric(N, N) + ∇_{N}(u^{−1}H) + u^{−1}|χ|_{γ}^{2},

where ∇^{2}u^{−1} denotes the Hessian of the function u^{−1}. On the other hand:

Δu^{−1} = ∇^{2}u^{−1}(N, N) + H ∇_{N}u^{−1} + Δ /_{γ}u^{−1}.

Combining the last two equations we obtain:

Δ /_{γ}u = 2u^{−1}|∇ /u|^{2} − u^{2}Δ /u^{−1} = −u Ric(N, N) − u∇_{N}H − u|χ|^{2}.

Substituting Ric(N, N) = \frac{1}{2}(H^{2} − |χ|^{2} + R − R /) from the Gauss equation, and the definitions H̄ = ruH, χ̄ = ruχ, and N̄ = ruN from the introduction, we get (11).

2. Bernoulli-type parabolic PDEs on 𝕊^{2}

In this section we collect the analytical results we need to prove the Main Theorem: (conditions for) global existence, uniqueness, asymptotic behavior and continuous dependence on parameters for solutions of (12). All these rely on a simple pointwise a priori bound whose proof we present here. Many of the results presented in this section are adapted from [2]. In order to ensure the uniform parabolicity of (12) we assume throughout this section that Γ = e^{−2v}/H̄ is bounded above and below by positive constants.

First, we define parabolic Banach spaces to be used in our study of (12). Let 0 < r_{0} < r_{1} ≤ ∞, I = [r_{0}, r_{1}] ⊂ ℝ^{+}, and let A_{I} = I × 𝕊^{2}. Given a function f on A_{I}, define:

[f]_{α;I} = sup_{\begin{matrix}(r_{1},θ_{1}),(r_{2},θ_{2})∈A_{I} \\ dist(θ_{1},θ_{2})<π \end{matrix}} [\frac{|f(r_{2}, θ_{2}) − f(r_{1}, θ_{1})|}{|1 − r_{2}/r_{1}|^{α/2} + dist(θ_{2}, θ_{1})^{α}} ] ,

∥f∥_{0;I} = sup_{A_{I}} |f|, ∥f∥_{0,α;I} = ∥f∥_{0;I} + [f]_{α;I}.

Here dist(·, ·) denotes the geodesic distance on 𝕊^{2}. If f(r, ·) is a tensor field on 𝕊^{2}, then f(r_{2}, θ_{2}) is understood to mean the parallel translate of f(r_{2}, θ_{2}) back to θ_{1} along the unique geodesic from θ_{1} to θ_{2}. With these conventions, we can now define:

∥f∥_{k,α;I} = ∑_{i+2j≤k} ∥∇ /^{i}(r∂_{r})^{j}f∥_{0;I} + ∑_{i+2j=k}[∇ /^{i}(r∂_{r})^{j}f]_{α;I},

where ∇ /^{i}f is to be interpreted as the i-th covariant derivative of f in the standard metric on 𝕊^{2}. Now define H_{I}^{k,α} to be the space of functions f on A_{I} for which ∥f∥_{k,α;I} is defined and finite. Equipped with the norm ∥·∥_{k,α;I}, the space H_{I}^{k,α} is a Banach space. Following Bartnik in [2], we also use the notation:

f^{∗}(r) = sup_{θ∈𝕊^{2}} f(r, θ), f_{∗}(r) = inf_{θ∈𝕊^{2}} f(r, θ).

2.1. Conditions for global existence and uniqueness. Our first observation is that equation (12) is uniformly parabolic with r as the ‘time’ variable. Therefore, given any initial data u(r_{0}, θ) = u_{0}(θ), it is standard to obtain the existence of a unique solution on a short time interval [r_{0}, r_{0} + ϵ) for some ϵ > 0. Furthermore, it is well known that, for some choices of coefficients and initial data, a classical solution can blow up in finite time. Thus, our main objective here is to derive conditions which guarantee the existence of a global positive solution on the time interval [r_{0}, ∞].

The principal ingredient in this and future subsections is a simple a priori bound on solutions of (12). To derive this bound, we use the familiar substitution w = u^{−2} well-known from the elementary method used to solve the corresponding Bernoulli ordinary differential equation. If u > 0 satisfies (12) on [r_{0}, r_{1}], then w satisfies

r∂_{r}w − β · ∇ /u = 2(−Γu^{−1}Δ /u − Aw + B). (15)

Since this equation is only used to derive pointwise a priori bounds, and since u has a maximum where w has a minimum and vice versa, there is no need to transform the gradient and Laplacian terms. For example, it follows from (15) that

r∂_{r}w_{∗} + 2A^{∗}w_{∗} ≥ 2B_{∗}, (16)

which upon integration yields the lower bound:

rw_{∗} ≥ [r_{0}w_{∗}(r_{0}) + ∫_{r_{0}}^{r} 2B_{∗} exp (∫_{r_{0}}^{r′}(2A^{∗} − 1)\frac{dt}{t}) dr′] exp (∫_{r_{0}}^{r}(1 − 2A^{∗})\frac{dt}{t}) . (17)

Similarly, one obtains the upper bound:

rw^{∗} ≤ [r_{0}w^{∗}(r_{0}) + ∫_{r_{0}}^{r} 2B^{∗}exp (∫_{r_{0}}^{r′}(2A_{∗} − 1)\frac{dt}{t}) dr′] exp (∫_{r_{0}}^{r}(1 − 2A_{∗})\frac{dt}{t}) . (18)

In particular it follows immediately that w is bounded above, and hence u ≥ c > 0, where c depends on r_{1}. Suppose furthermore that

K = \frac{1}{r_{0}} ( sup_{r_{0}<r<∞} (− ∫_{r_{0}}^{r} 2B_{∗} exp (∫_{r_{0}}^{s}(2A^{∗} − 1)\frac{dt}{t}) ds))^{−1/2} > 0. (19)

If the initial data u_{0} < K, then w is bounded below, and hence u < ∞. We note that if B ≥ 0 then K = ∞, which gives u < ∞ for any positive initial data. The above considerations lead to the following result:

Theorem 1. Let I = [r_{0}, ∞), suppose β, Γ, A, B ∈ C^{α}(I×𝕊^{2}), and suppose that the constant K defined in (19) is positive. Then, for any positive function u_{0} ∈ C(𝕊^{2}) satisfying sup_{𝕊^{2}} u_{0} < K, the equation (12) has a unique positive global classical solution u on I × 𝕊^{2} with initial data u(r_{0}, θ) = u_{0}(θ).

Now, suppose that the coefficients in (12) satisfy

∥β∥_{2,α;I′} + ∥Γ∥_{2,α;I′} + ∥A∥_{2,α;I′} + ∥B∥_{2,α;I′} ≤ C,

for some C > 0, and that u is a solution which is bounded above and below, C^{−1} ≤ u ≤ C on A_{I}, where I ⊂ I′. Then standard parabolic Schauder theory gives

∥u∥_{4,α;I} ≤ C′, (20)

where C′ depends on C and the length of I. Using the scaling properties of the H_{I}^{k,α}-norms and of equation (12) we can also derive (20) for I_{λ} = [λr_{0}, λr_{1}] with C′ independent of λ provided I_{λ} ⊂ I′. Furthermore, if in addition |2A − 1| < C/r on [r_{0}, ∞) × 𝕊^{2} and u(r_{0}, θ) < K, we may use the bounds (17) and (18) to obtain uniform infimum and supremum bounds on every subinterval of [r_{0}, ∞). Thus, we obtain the following result.

Theorem 2. Let I = [r_{0}, ∞), and suppose that there is a constant C > 0 such that β, Γ, A, B ∈ H_{I}^{2,α} satisfy

∥β∥_{2,α;I} , ∥Γ∥_{2,α;I} , ∥B∥_{2,α;I} , ∥r(2A − 1)∥_{2,α;I} ≤ C,
C^{−1} ≤ Γ ≤ C,

and u_{0} ∈ C^{4,α}(𝕊^{2}) satisfies 0 < u_{0} ≤ K − C^{−1}. Then there is a unique positive solution u ∈ H_{I}^{4,α} of (12) with initial condition u(r_{0}, θ) = u_{0}(θ). Furthermore, we have

∥u∥_{4,α;I} ≤ C′,

where C′ depends only on C.

Our final result concerning global existence is a simple consequence of the maximum principle.

Theorem 3. Let I = [r_{0}, ∞), and suppose that u > 0 is a classical solution of equation (12) on I × 𝕊^{2} with coefficients β, Γ, A, B ∈ H_{I}^{2,α} and with initial data u_{0} ∈ C^{4,α}(𝕊^{2}). Let B̃ ∈ H_{I}^{2,α} satisfy B̃ ≥ B. Then the equation

r∂_{r}u − β · ∇ /u = Γu^{2}Δ /u + Au − B̃u^{3}

has a unique solution ũ ∈ H_{I}^{4,α} with the same initial data ũ(r_{0}, θ) = u_{0}(θ). Furthermore, we have 0 < ũ ≤ u.

Proof. It suffices to prove a supremum a priori bound for ũ on any interval [r_{0}, r_{1}) where the solution ũ > 0 exists. Subtracting the equation for u from the equation for ũ, we get an equation for v = ũ − u:

r∂_{r}v − β · ∇ /v = Γũ^{2}Δ /v + Ãv − (B̃ − B)ũ^{3}, (21)

where Ã = A + Γ(ũ + u)Δ /u − B(ũ^{2} + ũu + ũ^{2}). Since (B̃ − B)ũ^{3} > 0, the maximum principle applies to give v ≤ 0. It follows that ũ ≤ u on [r_{0}, r_{1}).

For the remainder of Section 2 we set I = [r_{0}, ∞).

2.2. Asymptotic behavior. To study the asymptotic behavior of solutions of equation (12) we define:

m = \frac{r}{2} (1 − u^{−2}).

If u is a global solution of equation (12) on I × 𝕊^{2}, then m satisfies

r∂_{r}m − β · ∇ /m = rΓ\frac{Δ /u}{u} − (2A − 1)m + r(A − B). (22)

We note that if r |2A − 1| ≤ C, and

|A − B|^{∗} ∈ L^{1}(I), (23)

then using the maximum principle, it follows that |m| is bounded. Applying Schauder theory to equation (22) we then obtain:

Theorem 4. Let I = [r_{0}, ∞), let β, Γ, A, B ∈ H_{I}^{2,α}, suppose that (23) is satisfied, and let u ∈ H_{I}^{2,α} be a positive solution of equation (12). Suppose that there is a constant C > 0 such that

∥u∥_{2,α;I} , ∥r(2A − 1)∥_{2,α;I} , ∥B∥_{2,α;I}, ∥|A − B|^{∗}∥_{L^{1}(I)} ≤ C,
C^{−1} ≤ Γ ≤ C.

Then m = r(1 − u^{−2})/2 satisfies

∥m∥_{4,α;I} < C′, (24)

where C′ depends only on C.

We will also need the bound (24) in some cases where the condition (23) cannot be verified. To this end, we state a comparison result that establishes a bound on m. Let u_{i}, i = 1, 2, satisfy

r∂_{r}u_{i} − β · ∇ /u_{i} = Γu_{i}^{2}Δ /u_{i} + Au_{i} − B_{i}u_{i}^{3}. (25)

Setting m_{i} = r(1 − u_{i}^{−2})/2, i = 1, 2, and m̃ = m_{2} − m_{1}, we obtain, in view of (22),

r∂_{r}m̃ − β̃ · ∇ /m̃ = Γu_{2}^{2}Δ /m̃ + (2Ã − 1)m̃ + r(B_{1} − B_{2}),

where

β̃ = β + 3Γu_{2}^{4}∇ /(m_{1} + m_{2}),
Ã = A − Γu_{1}^{2}u_{2}^{2} (r^{−1}Δ /m_{1} + 3r^{−2}(u_{1}^{2} + u_{2}^{2}) |∇ /m_{1}|^{2}).

If we also assume that m_{1}, ∇ /m_{1}, Δ /m_{1}, u_{1}, and u_{2} are bounded, then r|2Ã − 1| is bounded, and provided that also |B_{1} − B_{2}|^{∗} ∈ L^{1}(I), we may apply the previous argument to derive a bound on |m̃|, and consequently also on |m_{2}|. This leads to the following theorem.

Theorem 5. Let I = [r_{0}, ∞), let β, Γ, A, B_{1}, B_{2} ∈ H_{I}^{2,α} and suppose that u_{i} > 0, i = 1, 2, are bounded classical solutions of equation (25) on I×𝕊^{2}. Let B = B_{1}−B_{2}, and suppose also that |B|^{∗} ∈ L^{1}(I), that m_{1} is bounded, and that there is a constant C > 0 such that

∥u_{1}∥_{2,α;I} , ∥u_{2}∥_{2,α;I} , ∥m_{1}∥_{2,α;I} , ∥r(2A − 1)∥_{2,α;I} , ∥B∥_{2,α;I}, ∥|B|^{∗}∥_{L^{1}(I)} ≤ C,

C^{−1} ≤ Γ ≤ C.

Then m̃ = r(1 − ũ^{−2})/2 satisfies

\left∥{ \atop m̃}\right∥_{4,α;I} ≤ C′,

where C′ depends only on C.

2.3. Continuous dependence on parameters.

Theorem 6. Let I = [r_{0}, ∞), and suppose that u_{λ} ∈ H_{I}^{k+2,α}, a ≤ λ ≤ b, is a family of solutions of (12) with β_{λ}, Γ_{λ}, A_{λ}, B_{λ} satisfying r(2A_{λ} − 1), r(2B_{λ} − 1), β_{λ}, Γ_{λ} ∈ C^{0}([a, b], H_{I}^{k,α}) and with the initial data u_{λ}(r_{0}) ∈ C^{0}([a, b], C^{k+2,α}(𝕊^{2})). Suppose also that one of the following conditions is satisfied:

1. |A_{λ} − B_{λ}|^{∗} ∈ C^{0}([a, b], L^{1}(I));

2. A continuous family of solutions m′_{λ} ∈ C^{0}([a, b], H_{I}^{k+2,α}) of (22) exist.

Then u_{λ}, m_{λ} ∈ C^{0}([a, b], H_{I}^{k,α}).

3. Deformation of metrics in ℳ

We are now in a position to sketch the proof of the Main Theorem. Recall that any metric g ∈ ℳ can be written as in (7). We define a nested sequence of subsets ℳ = ℳ_{0} ⊃ ⋯ ⊃ ℳ_{4}:

ℳ_{1} = {g ∈ ℳ_{0} : r(1 − u) ∈ H_{[r_{0},∞)}^{4,α}; rβ, rv ∈ H_{[r_{0},∞)}^{8,α}, ∀r_{0} > 0, supp R is compact},
ℳ_{2} = {g ∈ ℳ_{1} : 2κ − R > 0},
ℳ_{3} = {g ∈ ℳ_{2} : β, v are compactly supported},
ℳ_{4} = {g ∈ ℳ_{3} : g is flat}.

Let us say that ℳ_{i} is connected to ℳ_{i+1} if for each g ∈ ℳ_{i} there is a path Γ in ℳ_{i}, continuous in the topology of C_{−1}^{2,α}, with Γ(0) = g and Γ(1) ∈ ℳ_{i+1}. We will show that ℳ_{i} is connected to ℳ_{i+1} for each i = 0, …, 3. The Main Theorem follows by joining these paths.

Lemma 7. ℳ_{0} is connected to ℳ_{1}.

Proof. Let g = g_{0} ∈ ℳ_{0}. It is not difficult, using a truncation followed by a standard smoothing, to construct a deformation g_{λ}, continuous in C_{−1}^{2,α}, from g_{0} to a smooth metric g_{1} which is flat outside a large enough ball, with scalar curvature R_{λ} ∈ L^{1}, and such that g_{λ} − g_{0} is small in C_{−1/2}^{2,α} for all λ. Since g_{λ} is close to g, the coordinate spheres are still quasiconvex in g_{λ}, and the negative part R_{λ}^{−} of the scalar curvature of g_{λ} is small in L^{3/2}. It follows that the operator −8Δ_{g_{λ}} + R_{λ} is injective [4], and hence also a bijection; see [5, 12]. We can now choose a smooth positive function of compact support S_{λ} which is close to R_{λ} in C_{−5/2}^{α}, and solve the equation

(−8Δ + R_{λ})ψ_{λ} = R_{λ} − S_{λ}.

It follows from the above that ψ_{λ} is small in C_{−1/2}^{2,α}. Taking ϕ_{λ} = 1 + ψ_{λ}, we see that the metrics ϕ_{λ}^{4}g_{λ} have positive scalar curvature, quasiconvex coordinate spheres, and form a continuous path from g_{0} to a smooth metric g̃_{1} = ϕ_{1}^{4}g_{1}. Since R_{1} − S_{1} is of compact support, and g_{1} is flat outside a compact set, it follows that g̃_{1} ∈ C_{−1}^{k,α} for all k. On each coordinate sphere equipped with the metric γ induced by g̃_{1}, it is possible, using the techniques of [8, Chapter 2], to find a uniformization factor r^{2}e^{2v}, with bounds as required in ℳ_{1}, so that γ̄ = r^{−2}e^{−2v}γ is a round metric with surface area 4πr^{2}. We conclude that the continuous path ϕ_{λ}^{4}g_{λ} joins g ∈ ℳ_{0} to a metric g̃_{1} ∈ ℳ_{1}.

It is important to note that since the round metric γ̄ on the coordinate spheres will in general vary with r, it is most likely necessary to change the background flat metric when writing g̃_{1} as in (7). Nevertheless, these two flat metrics are asymptotic as r → ∞; see [13] for details.

As outlined in the Introduction, the deformation is obtained in the next three steps by deforming g_{λ} explicitly inside a ball B_{r_{0}}, while solving (12) outside B_{r_{0}} with the deformation of β_{λ}, v_{λ} and R_{λ} defined so that κ_{λ}, H̄_{λ} > 0, R_{λ} ≥ 0, and so that theorems from Section 2 guarantee global existence, asymptotic behavior as r → ∞, and continuity of u_{λ} in H_{[r_{0},∞)}^{4,α}. Note that in order to ensure continuity at the end point of the deformation, it is necessary to have a H̄_{λ} uniformly bounded below by a positive constant. Now, if g_{λ} is a path in ℳ_{i}, i = 1, 2, 3, such that for some 0 < r′ < r_{0}, g_{λ}|_{B_{r_{0}}} is continuous in C^{2}, and r(1 − u_{λ}), rβ_{λ}, and rv_{λ} are continuous in H_{[r′,∞)}^{4,α}, then g_{λ} is continuous in C_{−1}^{2,α}.

Lemma 8. ℳ_{1} is connected to ℳ_{2}.

Proof. Let g ∈ ℳ_{1}, choose r_{1} such that 2κ − R > 0 for r < r_{1}, and r_{0} with r_{0} < r_{1}. For each λ ∈ [0, 1], define g_{λ} = g inside B_{r_{0}}, β_{λ} = β, and v_{λ} = v everywhere. We have κ_{λ}, H̄_{λ} > 0. Let φ(r) be a smooth cut-off function on [0, ∞), satisfying 0 ≤ φ ≤ 1, φ = 1 on [0, r_{0}], and φ = 0 on [r_{1}, ∞). Define φ_{λ}(r) = (1 − λ) + λφ(r) and define R_{λ} = φ_{λ}R. Then R_{λ} is monotonically decreasing in λ, R_{λ} = R on B_{r_{0}}, and supp(R_{1}) ⊂ B_{r_{1}}. Thus, Theorems 3 and 5 can be used to solve equation (12) on [r_{0}, ∞) × 𝕊^{2} for u_{λ} ∈ H^{4,α}. The continuity of u_{λ} and m_{λ} with respect to λ is obtained from Theorem 6. Clearly, g_{1} ∈ ℳ_{2} and the lemma follows.

Lemma 9. ℳ_{2} is connected to ℳ_{3}.

Proof. Let g ∈ ℳ_{2}, and put R_{λ} = R. For λ ∈ [1, ∞) define β̃_{λ} = (ϕ_{λ})^{∗}β, ṽ_{λ} = (ϕ_{λ})^{∗}v, where ϕ_{λ}(r, θ) = (λr, θ). Note that rβ̃_{λ} and rṽ_{λ} are continuous in H_{I}^{6,α} since rβ̃_{λ} and rṽ_{λ} are uniformly bounded in H_{I}^{8,α}. Now, let β_{λ} = φβ + (1 − φ)β̃_{λ}, and define v_{λ} by e^{2v_{λ}} = φe^{2v} + (1 − φ)e^{2ṽ_{λ}}, where φ(r) is a cut-off function as in the proof of the previous lemma. It follows from (9) that

e^{2v_{λ}} H̄_{λ} = φe^{2v}H̄ + (1 − φ)e^{2ṽ_{λ}} H̃_{λ} + (e^{2v} − e^{2ṽ_{λ}})rφ′,

where H̃_{λ} = (ϕ_{λ})^{∗}H̄. Thus, since v and ṽ_{λ} tend to zero as r → ∞, it follows that if r_{0} and r_{1}/r_{0} are large enough, then H̄_{λ} > 0 for r > r_{0}. Furthermore, in view of (10), the Gauss curvature κ_{λ} is given by

r^{2}e^{2v_{λ}}κ_{λ} = 1 − Δ /v_{λ} = r^{2}(φe^{2v}κ + (1 − φ)e^{2ṽ_{λ}} κ̃_{λ}) + |∇ /v|^{2}

− 2e^{−2v_{λ}}(φe^{2v} |∇ /v|^{2} + (1 − φ)e^{2ṽ_{λ}} |∇ /ṽ_{λ}|^{2}),

where κ̃_{λ} = (ϕ_{λ})^{∗}κ. Hence, since also |∇ /v| and |∇ /ṽ_{λ}| tend to zero as r → ∞, we see that if r_{0} is large enough, then κ_{λ} > 0 for r > r_{0}. By choosing r_{0} large enough, we can also ensure that R = 0 outside B_{r_{0}}. As in the proof of the previous lemma, we define g_{λ} = g inside B_{r_{0}}, and solve equation (12) for u_{λ} outside B_{r_{0}}. The existence of u_{λ} for all r ≥ r_{0} is now guaranteed by Theorem 1. Note that outside B_{r_{1}}, β_{λ} = β̃_{λ}, v_{λ} = ṽ_{λ}, hence we have a uniformly bounded solution λ^{−1}(ϕ_{λ})^{∗}m of equation (22), and therefore Theorem 5 applies to give the asymptotic behavior of u_{λ} for r → ∞. It is easy to see that the path g_{λ} can be extended continuously to [1, ∞], and since β_{λ} and v_{λ} tend to zero as λ → ∞ for r > r_{1}, it follows that g_{∞} ∈ ℳ_{3}.

Lemma 10. ℳ_{3} is connected to ℳ_{4}.

Proof. Let g ∈ ℳ_{3}, choose r_{0} > 0 so that R, β, and v are supported in B_{r_{0}}, and let φ(r) be a cut-off function as above. For λ ∈ [1, ∞), define β_{λ} = φ (ϕ_{1/λ})^{∗}β, and ṽ_{λ} = (ϕ_{1/λ})^{∗}v. Let r_{0} < r_{1} < r_{2}, and let ζ(r) be a smooth function satisfying 0 ≤ ζ ≤ 1, ζ = 1 on [0, r_{1}], ζ = 0 on [r_{2}, ∞). Let H̃_{λ} = (ϕ_{1/λ})^{∗}H̄, and note that inf H̃_{λ} = inf H̄ > 0, hence h = inf H̃_{λ} is independent of λ. Let f(r) be a smooth non-negative function supported on [r_{0}, r_{2}], satisfying on [r_{0}, r_{1}] the inequality:

f > −r^{−a−1}(φh + rφ′),

where a = max{−(2 + 2r∂_{r}ṽ_{λ}), 0}. Let ξ(r) = r^{a} ∫_{r_{0}}^{r} f(s) ds, ψ = ξ + φ, and v_{λ} = ζ(ṽ_{λ} + \frac{1}{2} log ψ). Since ξ ≥ 0, it now follows from (9) that we have for r_{0} < r < r_{1}:

ψH̄_{λ} = ξ(2 + 2r∂_{r}ṽ_{λ}) + rξ′ + φH̃_{λ} + rφ′ > −aξ + rξ′ − r^{a+1}f = 0.

Furthermore, since β_{λ} = 0 in B_{r_{2}} \ B_{r_{1}}, we can also choose ζ so as to ensure that H̄_{λ} > 0 there, provided that r_{2}/r_{1} is large enough. Since the deformation of v is radial, it is clear that κ_{λ} > 0. Define g_{λ} = λ^{2}(ϕ_{1/λ})^{∗}g in B_{r_{0}}, and as before, solve for u_{λ} in (12) on [r_{0}, ∞) × 𝕊^{2} with initial data u_{λ}|_{S_{r_{0}}} . Global existence and asymptotic behavior as r → ∞ is obtained from Theorems 1 and 4. The path g_{λ} can be extended continuously to [1, ∞], and since β_{λ}, v_{λ}, and R_{λ} tend to zero as λ → ∞, it follows that u_{λ} tends to 1. Consequently g_{∞} is flat, and the continuous path g_{λ} joins g_{1} to a flat metric g_{∞} ∈ ℳ_{4}. However, note that g_{1} ≠ g, since clearly v_{1} ≠ v. In order to complete the proof of the lemma, we now define a continuous path g_{λ}, λ ∈ [0, 1], between g and g_{1}. Define g_{λ} = g in B_{r_{0}}, β_{λ} = β, R_{λ} = R, and v_{λ} = ζ(v + (1/2) log(1 − λ + λψ)). Then from (9) we get e^{2v_{λ}} H̄_{λ} = (1 − λ)e^{2v_{0}} H̄_{0} + λe^{2v_{1}} H̄_{1} > 0 in [r_{0}, r_{1}], and as before H̄_{λ} > 0 also in [r_{1}, r_{2}] provided r_{2}/r_{1} is large enough. Clearly, as before, we have κ_{λ} > 0. Thus, we can solve for u_{λ} in (12) as above.

Note that due to the uniformization step in Lemma 7, the final flat metric g_{∞} in Lemma 10 may be different from the background flat metric originally given in ℳ_{0}.

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University of Alabama at Birmingham, Birmingham, AL 35205

E-mail address: smith@math.uab.edu

University of Alabama at Birmingham, Birmingham, AL 35205

E-mail address: weinstei@math.uab.edu