LOCAL RIGIDITY OF SURFACES IN SPACE FORMS

MICHAEL T. ANDERSON

Abstract. We prove that isometric embeddings of closed, embedded surfaces in ℝ^{3} are locally rigid, i.e. they admit no non-trivial local isometric deformations, answering a question in classical differential geometry. The same result holds for abstract surfaces embedded in constant curvature 3-manifolds, provided a mild condition on the fundamental group holds.

1. Introduction.

A well-known question in classical differential geometry is whether isometric embeddings of closed surfaces Σ in Euclidean space ℝ^{3} are locally rigid. Thus, suppose γ is a smooth Riemannian metric on Σ and

F : (Σ, γ) ↪ (ℝ^{3}, g_{0})

is a smooth isometric embedding, where g_{0} is the Euclidean metric. If F_{s} is a curve of isometric embeddings of Σ in ℝ^{3} with F_{0} = F, does F_{s} necessarily arise from a curve σ_{s} in the group of isometries of ℝ^{3}, i.e. F_{s} = σ_{s} ∘ F_{0}? Of course open surfaces such as a flat plane, are not locally rigid. Thus, throughout the paper, we assume that Σ is compact and connected; we also asume that the curve F_{s} is C^{1} in s.

It is useful to first describe some background setting and results. In case Σ is strictly convex, or equivalently when the Gauss curvature K_{γ} of γ is positive, it is well known that (Σ, γ) is locally rigid. In fact surfaces of strictly positive curvature are globally rigid, in that isometric embeddings are uniquely determined up to rigid motion of ℝ^{3}. Classical proofs of this fact have been given by Liebmann and Cohn-Vossen.

On the other hand, in general, closed surfaces are not globally rigid; there exist isometric but non-congruent surfaces in ℝ^{3}. A well-known class of examples is as follows, cf. [17]. Consider a C^{∞} closed surface Σ_{0} ⊂ ℝ^{3} which contains a flat, planar domain, i.e. a domain U ⊂ P, where P is a plane in ℝ^{3}. Let Σ_{1} be a surface obtained by locally smoothly perturbing Σ_{0} within U to one side of U, and let Σ_{2} be the surface equal to Σ_{1} outside U, while inside Σ_{2} is the reflection of Σ_{1} through P. Then Σ_{1} and Σ_{2} are isometric, (since reflection is an isometry of ℝ^{3}), but noncongruent surfaces. Moreover, one may choose an initial smooth surface Σ_{0} which has an infinite number of such planar regions and by reflection in each, produce an infinite number of distinct isometric but non-congruent C^{∞} surfaces Σ_{i} ⊂ ℝ^{3} converging smoothly to a limit surface Σ ⊂ ℝ^{3}. This construction can be extended to give uncountably many, (for instance Cantor-like sets), of isometric but non-congruent surfaces, cf. [15].

In addition, carrying out the construction above infinitesimally shows that any locally planar surface Σ_{0} as above has non-trivial infinitesimal isometric deformations, i.e. deformations preserving the induced metric γ to first order which do not arise from the restriction of Killing fields on ℝ^{3}, cf. also Remark 2.7. We refer to the text by Spivak [17] for more detailed background on these rigidity questions and to [14] and [18] for more recent commentary and references.

In this work, we answer the local rigidity question above affirmatively:

Partially supported by NSF Grant DMS 0604735.

Theorem 1.1. Let γ be a C^{m,α} Riemannian metric on a closed connected surface Σ, m ≥ 3. Then any isometric C^{m+1,α} embedding of Σ in ℝ^{3} is C^{1} locally rigid.

The proof of Theorem 1.1 also gives the local isometric rigidity of closed embedded surfaces in the 3-dimensional spaces of constant curvature, 𝕊^{3}, ℝ^{3}, ℍ^{3}. We also consider the following slightly more abstract situation. Let (M^{3}, g) be any compact, oriented, constant curvature 3-manifold with non-empty boundary ∂M, and suppose that

(1.1)  π_{1}(M, ∂M) = 0.

The condition (1.1) is equivalent to the statement that ∂M is connected and the inclusion map induces a surjection π_{1}(∂M) → π_{1}(M); in turn the latter condition is equivalent to the statement that the lift of ∂M is connected in all covering spaces M̂ of M. The definition of local rigidity in this context is the same as above, namely rigidity of local curves of isometric embeddings of (∂M, γ) into (M, g) near a given isometric embedding, modulo isometries of (M, g).

Theorem 1.2. Let (M^{3}, g) be a compact oriented 3-manifold with metric g of constant curvature with non-empty boundary (∂M, γ), and suppose g is C^{m,α}, m ≥ 3, up to ∂M inducing a C^{m,α} boundary metric γ on ∂M.

If (1.1) holds, then (∂M, γ) is C^{1} locally rigid in (M, g).

Theorem 1.2 is of course analogous to Theorem 1.1 with Σ = ∂M and M ⊂ ℝ^{3} equal to the compact region bounded by Σ. The condition (1.1) is not necessary in Theorem 1.1, due to the fact that here the domain M is embedded in the complete, simply connected space ℝ^{3}. On the other hand, Theorem 1.2 generalizes Theorem 1.1 to a class of so-called Alexandrov immersed surfaces in ℝ^{3}, i.e. isometric immersions Σ → ℝ^{3} which extend to an immersion of a 3-manifold M with ∂M = Σ and satisfying the handlebody-type condition (1.1).

Next consider infinitesimal analogues of these results. A vector field X defined along ∂M is called an infinitesimal isometry at ∂M if L_{X}g = 0 when restricted to T (∂M); such vector fields are also often called infinitesimal bendings or infinitesimal isometric deformations. A surface (∂M, γ) is C^{k,β} infinitesimally rigid in (M, g) if any C^{k,β} infinitesimal isometry X at ∂M extends to a Killing field on (M, g).

As noted above, surfaces in space-forms are not always infinitesimally rigid. However, we prove that infinitesimally bendable surfaces for which the bending vector field X is sufficiently smooth are very special; they are necessarily planar, i.e. the 2^{nd} fundamental form A vanishes on an open set. Thus, define a surface Σ in a 3-manifold of constant curvature to be non-planar if A does not vanish on any open set of ∂M.

Theorem 1.3. Suppose (Σ, γ) is a closed, C^{m+1,α} embedded surface in one of the 3-dimensional space-forms 𝕊^{3}, ℝ^{3}, ℍ^{3}, or (Σ, γ) = (∂M, γ), where (M, g) is of constant curvature with metric C^{m,α} up to ∂M, m ≥ 3, and suppose (1.1) holds.

If (Σ, γ) is non-planar, then (Σ, γ) is C^{m+1,α} infinitesimally rigid.

It is interesting to note that this result is almost sharp, in that by a result of Cohn-Vossen [7], there exist C^{∞} smooth surfaces of revolution in ℝ^{3} which have C^{2} infinitesimal bendings, and so are not C^{2} infinitesimally rigid, cf. also [17]. Note also that some hypothesis such as (1.1) is necessary in both Theorems 1.2 and 1.3, since without it there are numerous counterexamples. For example, let M = S^{3} \ (B_{1} ∪ B_{2}), where B_{i} are a pair of disjoint round 3-balls in S^{3} endowed with a round metric. Then a generic pair of Killing fields X_{i} on S_{i}^{2} = ∂B_{i} does not extend to a Killing field on M. To illustrate the situation where π_{1}(∂M) does not surject onto π_{1}(M), let for instance M = T^{3} \ B, where B is a round 3-ball in a flat 3-torus T^{3}. Then again Killing fields on ∂M do not extend to Killing fields on T^{3}.

The proofs of Theorems 1.1 and 1.2 are essentially the same, and in the main proceed by first proving Theorem 1.3. Local rigidity follows easily from infinitesimal rigidity by integration, and, via the proof of Theorem 1.3, the proof of local rigidity for curves of planar surfaces is basically elementary. Thus, the main point is to prove Theorem 1.3.

The method of proof of Theorem 1.3 is to examine global properties of the space of constant curvature, or equivalently, Einstein metrics g on M. Thus, the moduli space E of such metrics is a smooth Banach manifold, for which the map to the boundary metrics (Dirichlet boundary map),

(1.2)  Π : E → Met(∂M),

is C^{∞} smooth, cf. Theorem 2.1. As shown in §2, (cf. Remark 2.9), Theorem 1.3 is then quite easy to prove when the metric (M, g) is non-degenerate, in the strong sense that the derivative DΠ of Π at g has trivial kernel. (On the other hand, KerDΠ is always infinite dimensional on planar surfaces).

The main work involves an understanding of the cokernel of DΠ in Met(∂M). Although DΠ may not be surjective in general, we show that, roughly speaking, given any global constant curvature metric g on M, any infinitesimal deformation h ∈ T_{γ}Met(∂M) of the boundary metric γ is realised by an infinitesimal constant curvature deformation of g defined (only) near ∂M. (This is somewhat of an over-simplification; see Proposition 2.16 for the exact statement). This “local” surjectivity, (i.e. surjectivity for metrics defined only near ∂M), is then combined with a key identity, cf. Proposition 2.13, for the Lie derivative of divergence-free symmetric forms with respect to infinitesimal isometries at the boundary (∂M, γ). This allows one to prove that any infinitesimal isometry at (∂M, γ) also preserves the second fundamental form A of (∂M, γ) in (M, g), and the main result follows from a suitable unique continuation theorem for constant curvature metrics on M, analogous to the fundamental theorem for surfaces in ℝ^{3}.

The constraint equations induced on the data (∂M, γ, A) play a very important rôle throughout the proof. Thus, the constraint induced by the Gauss-Codazzi equations is closely related to the identity in Proposition 2.13. The constraint induced by the Gauss equations, (Gauss’ Theorema Egregium), which is closely related to well-known loss-of-derivative issues for isometric embeddings, is important in understanding regularity and Fredholm properties of the boundary map Π in (1.2).

In §2, we develop the main ideas in the proof and establish most of the main results needed to carry out the proof. Two of the main results, Theorems 2.1 and 2.2, are then proved, (independently of the results in §2), in §3.

I would like to thank Jean-Marc Schlenker, Mohammad Ghomi and Victor Alexandrov for very useful remarks and questions regarding a previous version of this work.

2. The Approach and Main Results

Throughout the paper, M denotes a connected, compact, oriented 3-dimensional manifold with compact, non-empty boundary ∂M.

Consider the Banach space

(2.1)  Met(M) = Met^{m,α}(M)

of Riemannian metrics on M which are C^{m,α} smooth up to ∂M. Here m is any fixed integer with m ≥ 2, including m = ∞, and α ∈ (0, 1). Let

(2.2)  𝔼 = 𝔼^{m,α}(M) ⊂ Met^{m,α}(M)

be the subset of Einstein metrics on M, C^{m,α} smooth up to ∂M, with

(2.3)  Ric_{g} = λg,

for λ arbitrary, but fixed. Of course this is the same as the space of constant curvature metrics, of curvature c = λ/2. We focus here on Einstein metrics since the equations (2.3) are formally a determined system of equations, and also because most of the results of generalize to Einstein metrics in higher dimensions, cf. Remark 3.5. In the following, the smoothness index (m, α) will sometimes be suppressed from the notation when its exact value is unimportant.

The space 𝔼^{m,α}(M) ⊂ Met^{m,α}(M) is invariant under the action of the group D_{1} = D_{1}^{m+1,α} of orientation preserving C^{m+1,α} diffeomorphisms of M equal to the identity on ∂M. This action is free, (since any such isometry equal to the identity on ∂M is necessarily the identity), and well-known to be proper. The moduli space E = E^{m,α}(M, λ) of λ-Einstein (or constant curvature) metrics on M is defined to be the quotient

(2.4)  E = 𝔼/D_{1}.

One has a natural Dirichlet boundary map

(2.5)  Π : 𝔼 → Met(∂M); Π(g) = γ = g|_{T (∂M)}.

which clearly descends to a map

(2.6)  Π : E → Met(∂M); Π([g]) = γ.

The starting point is the following result.

Theorem 2.1. Suppose either π_{1}(M, ∂M) = 0 or M is a domain in the simply connected form M_{c}^{3} of constant curvature c, (so that M_{c}^{3} is either 𝕊^{3}, ℝ^{3}, or ℍ^{3} up to rescaling), and ∂M is connected.

Then the space E is a C^{∞} smooth Banach manifold, (Fréchet manifold when m = ∞), and the boundary map Π is C^{∞} smooth.

Theorem 2.1 is proved by a suitable application of the implicit function theorem; the proof is given in §3. Strictly speaking this result is not needed for the proof of Theorems 1.1−1.3; it suffices to work at the formal, linearized level. However, the methods used to prove Theorems 1.1 − 1.3 easily give a proof of Theorem 2.1. This result also places the arguments to follow in a natural setting.

It is not clear to what extent Theorem 2.1 remains valid in general, when the assumption π_{1}(M, ∂M) = 0 is removed, cf. Remark 3.4. Note also that the examples discussed in §1 show that the boundary map Π is sometimes far from being 1 − 1, even locally.

Consider the Einstein operator

(2.7)  E : Met(M) → S_{2}(M),
E(g) = Ric_{g} − λg,

where S_{2}(M) is the space of symmetric bilinear forms on M. The linearization of E is given by

(2.8)  L_{E}(k) = 2\frac{d}{dt}(Ric_{g+tk} − λ(g + tk))|_{t=0} = D^{∗}Dk − 2R(k) − 2δ^{∗}β(k);

here δ^{∗}X = \frac{1}{2}L_{X}g, β(k) = δ(k) + \frac{1}{2}dtrk is the Bianchi operator with respect to g and R(h) is the action of the curvature tensor on symmetric bilinear forms k, cf. [4] for instance.

The tangent space T_{g}𝔼 equals KerL_{E} and the derivative of the Dirichlet boundary map Π in (2.5) is given, for L_{E}(k) = 0, by

(2.9)  DΠ_{g}(k) = k^{T} |_{∂M},

where k^{T} is the tangential projection or restriction of k to T (∂M). Thus k^{T} is the variation of the boundary metric γ = Π(g). It will also be important to consider the variation of the 2^{nd} fundamental form A of ∂M in M. Thus, analogous to (2.6), one has a natural Neumann boundary map

(2.10)  Π_{N} : E → S^{2}(∂M), Π_{N}([g]) = A.

This is well-defined, since A is invariant under the action of D_{1}. Note also that Π_{N} maps E^{m,α} to S_{2}^{m−1,α}(∂M). To compute the variation or derivative of Π_{N}, let g_{s} = g + sk be a variation of g. Since A = \frac{1}{2}L_{N}g, one has 2A′_{k} ≡ 2\frac{d}{ds}A_{g_{s}}|_{s=0} = (L_{N}g)′ = L_{N}k + L_{N′}g. A simple computation gives N′ = −k(N)^{T} − \frac{1}{2}k_{00}N, where k(N)^{T} is the component of k(N) tangent to ∂M and k_{00} = k(N, N). Thus

(2.11)  2A′_{k} = L_{N}k + δ^{∗}V,

where V = 2N′.

The kernel of DΠ in (2.5) or (2.9) consists of forms k satisfying L_{E}(k) = 0 and k^{T} = 0 on ∂M, while the kernel of DΠ_{N} in (2.10) consists of such forms satisfying (A′_{k})^{T} = 0 at ∂M. Thus, if both conditions hold,

(2.12)  k^{T} = 0, (A′_{k})^{T} = 0 at ∂M,

then k is both Dirichlet and Neumann degenerate. We note that each of the conditions in (2.12) is gauge-invariant, i.e. invariant under the addition of terms of the form δ^{∗}Z with Z = 0 on ∂M. Of course any form k satisfying k = ∇_{N}k = 0 at ∂M satisfies (2.12). Changing such k by arbitrary such gauge transformations shows that (2.12) is equivalent to the statement that k is pure gauge, to first order at ∂M, i.e.

(2.13)  k = δ^{∗}Z + O(t^{2}),

near ∂M, with Z = 0 on ∂M, where t(x) = dist_{g}(x, ∂M).

Theorem 2.2. Let g be an Einstein, i.e. constant curvature, metric on M, C^{m,α}, m ≥ 2, up to ∂M and suppose k is an infinitesimal Einstein deformation which is both Dirichlet and Neumann degenerate, so that L_{E}(k) = 0 and (2.12) holds. If k ∈ S_{2}^{2}(M) satisfies the Bianchi gauge condition β(k) = 0, and either π_{1}(M, ∂M) = 0 or M ⊂ M_{c}^{3} with ∂M connected, then

(2.14)  k = 0 on M.

The proof of Theorem 2.2 is also given in §3. This result is a (linearized) analogue, in a gaugefixed setting, of the well-known fundamental theorem for surfaces in ℝ^{3}: an isometric embedding of a surface Σ in ℝ^{3} is uniquely determined, up to congruence, by the pair (γ, A), cf. [10], [17].

The basic approach in the proof of Theorem 1.3 is to show that if an infinitesimal deformation k = δ^{∗}X of (M, g) is in Bianchi gauge and preserves the boundary metric γ to first order, i.e.

(2.15)  (L_{X}g)^{T} = 2(δ^{∗}X)^{T} = 0 at ∂M,

so that X is an infinitesimal isometry at ∂M, then the deformation k necessarily also preserves A tangentially to first order, when (∂M, γ) is non-planar, so that

(2.16)  (L_{X}A)^{T} = (A′_{k})^{T} = 0 at ∂M,

cf. (2.73). Given this, Theorem 1.3 then follows easily from Theorem 2.2. Theorem 2.2 is also an important part of the proof of Theorem 2.1.

The proof that (2.16) holds for sufficiently smooth infinitesimal isometries of non-planar surfaces begins with Proposition 2.13 below; however, in order to apply this result effectively, we need to establish a fair number of preliminary results along the way, (some of independent interest). At a first reading, one might want to jump at this point to Proposition 2.13 and the discussion following it, and refer back to the intervening steps as needed.

As is well-known, the operator E is not elliptic, due its covariance under diffeomorphisms: thus L_{E}(δ^{∗}Y) = 0, for any vector field Y on M, (at an Einstein metric). The proof of Theorem 1.3, (and Theorem 2.1), requires ellipticity at several points and so one needs a choice of gauge to break the diffeomorphism invariance of the Einstein equations. In view of (2.8), the simplest and most natural choice for the work to follow is the Bianchi gauge. Thus, let g̃ be a fixed (background) metric in 𝔼^{m,α}. The associated Bianchi-gauged Einstein operator is given by the C^{∞} smooth map

(2.17)  Φg̃ : Met^{m,α}(M) → S_{2}^{m−2,α}(M),
Φ(g) = Φg̃(g) = Ric_{g} − λg + δ_{g}^{∗}βg̃(g),

where βg̃(g) = δg̃g + \frac{1}{2}dtrg̃g is the Bianchi operator with respect to g̃, while δ^{∗} is taken with respect to g. Although Φg̃ is defined for all g ∈ Met^{m,α}(M), we will only consider it acting on g near g̃.

The linearization of Φ at g̃ = g is given by

(2.18)  L : Tg̃Met(M) → S_{2}(M),
L(h) = 2(DΦ)g̃(h) = D^{∗}Dh − 2R(h).

The operator L is formally self-adjoint and is clearly elliptic. Comparing (2.7) and (2.17), the relation between L and the linearization L_{E} = E′ of the Einstein operator E in (2.8) is given by

(2.19)  L_{E} = L − 2δ^{∗}β.

Clearly g ∈ 𝔼 if Φg̃(g) = 0 and βg̃(g) = 0, so that g is in the Bianchi gauge with respect to g̃. Given g̃, let Met_{B}(M) = Met_{B}^{m,α}(M) be the space of C^{m,α} smooth Riemannian metrics on M in Bianchi gauge with respect to g̃ at ∂M:

(2.20)  Met_{B}(M) = {g ∈ Met(M) : βg̃(g) = 0 at ∂M}.

Let

(2.21)  Z_{B} = {g ∈ Met_{B}(M) : Φ(g) = 0}

and let 𝔼_{B} ⊂ Z_{B} be the subset of Einstein metrics g in Z_{B}.

To justify the use of Φ, we need to show that the opposite inclusion holds, so that 𝔼_{B} = Z_{B}. This is done next in Lemmas 2.3 - 2.6. Thus, let χ_{1}^{k,α} be the space of C^{k,α} vector fields on M which vanish on ∂M, so that χ_{1}^{k,α} is the formal tangent space to D_{1}^{k,α} at the identity. For g ∈ Z_{B}, one then has

βg̃(g) = V ∈ χ_{1}^{m−1,α},

and one needs to show that δ^{∗}V = 0. (Here and below we identify vector fields and 1-forms via the metric g). This will require several elementary Lemmas, which will also be of importance later.

Lemma 2.3. For g in Met^{m,α}(M), one has

(2.22)  T_{g}Met^{m−2,α}(M) ≃ S_{2}^{m−2,α}(M) = Kerδ ⊕ Imδ^{∗},

where δ^{∗} acts on χ_{1}^{m−1,α}.

Proof: We first observe that the operator δδ^{∗} : χ_{1}^{m−1,α} → χ^{m−3,α} is an isomorphism. To see this, if X ∈ Kerδδ^{∗}, then

0 = ∫_{M}⟨δδ^{∗}X, X⟩ = ∫_{M} |δ^{∗}X|^{2} − ∫_{∂M}(δ^{∗}X)(X, N).

Since X = 0 on ∂M, it follows that δ^{∗}X = 0, so that X is a Killing field on M, vanishing on ∂M. Hence X = 0 on M so that δδ^{∗} is injective. Since the operator δδ^{∗} is self-adjoint with this boundary condition, it follows that it is also surjective, and hence an isomorphism. By elliptic regularity [9], [11], the inverse map is bounded.

To obtain the splitting (2.22), given h ∈ S_{2}^{m−2,α}, consider the equation δδ^{∗}X = δh ∈ C^{m−3,α}. If X = 0 at ∂M, this has a unique solution X with X ∈ χ_{1}^{m−1,α}. Setting π = h − δ^{∗}X gives the result.

Lemma 2.4. For g̃ ∈ 𝔼^{m,α} and g in Met^{m,α}(M) close to g̃, one has

(2.23)  T_{g}Met^{m−2,α}(M) ≃ S_{2}^{m−2,α}(M) = Kerβ ⊕ Imδ^{∗},

where β is the Bianchi operator with respect to g̃.

Proof: By the same argument as in Lemma 2.3, it suffices to prove that the operator βδ^{∗} : χ_{1}^{m−1,α} → χ^{m−3,α} is an isomorphism. Since this is an open condition, it suffices to prove this when g = g̃ is Einstein. A standard Weitzenbock formula gives

2βδ^{∗}X = D^{∗}DX − Ric(X) = D^{∗}DX − λX.

Hence, if λ ≤ 0, βδ^{∗} is a positive operator and it follows easily that βδ^{∗} is an isomorphism, (as in the proof of Lemma 2.3).

When λ > 0, this requires some further work. First, note that β itself is surjective. To see this, suppose Y is a 1-form (or vector field) orthogonal to Imβ. Then

(2.24)  0 = ∫_{M}⟨β(h), Y ⟩ = ∫_{M}⟨h, β^{∗}Y ⟩ − ∫_{∂M}[h(N, Y) − \frac{1}{2}trh⟨Y, N⟩].

Since h is arbitrary, this implies β^{∗}Y = δ^{∗}Y + \frac{1}{2}δY g = 0, and hence δ^{∗}Y = 0. The boundary term also vanishes, which implies Y = 0 at ∂M, and hence, as in the proof of Lemma 2.3, Y = 0, which proves the claim.

To prove that βδ^{∗} is surjective, it then suffices to show that for any h ∈ S_{2}^{m−2,α}(M), there exists X such that βδ^{∗}(X) = β(h). Via (2.22), write h = k +δ^{∗}Y with δk = 0. Then β(h) = \frac{1}{2}dtrk +βδ^{∗}(Y). This shows that it suffices to prove βδ^{∗} is surjective onto exact 1-forms df.

Thus, suppose there exists f such that df ⊥ Imβδ^{∗} = Im(D^{∗}D − λI). Arguing just as in (2.24), it follows that −dΔf − λ(df) = 0 on M, with boundary condition df = 0 at ∂M. Hence, Δf + λf = const, with f = const and N(f) = 0 at ∂M. It then follows from unique continuation for Laplace-type operators that f = const on M, and hence βδ^{∗} is surjective.

To see that βδ^{∗} = D^{∗}D − λI is injective, the family D^{∗}D − sλI for s ∈ [0, 1], with boundary condition X = 0 on ∂M, is a curve of elliptic boundary value problems. Since the index is 0 when s = 0, it follows that the index is also 0 when s = 1, i.e. βδ^{∗} has index 0 on χ_{1}, which proves the injectivity. This completes the proof.

Corollary 2.5. Any metric g ∈ Z_{B} near g̃ is necessarily Einstein, with Ric_{g} = λg, and in Bianchi gauge with respect to g̃, i.e.

(2.25)  βg̃(g) = 0.

Similarly, if k ∈ Met_{B}(M) is an infinitesimal deformation of g̃ in Z_{B}, i.e. L(k) = 0, then k is an infinitesimal Einstein deformation and β(k) = 0.

Proof: Since g ∈ Z_{B}, one has Φ(g) = 0, i.e.

Ric_{g} − λg + δ_{g}^{∗}βg̃(g) = 0.

The Bianchi identity β_{g}(Ric_{g}) = 0 implies

β_{g}(δ_{g}^{∗}(V)) = 0,

where V = βg̃(g). By the constraint (2.20), the vector field V vanishes on ∂M, so that V ∈ χ_{1}^{m−1,α}. It then follows from Lemma 2.4 that

(2.26)  δ^{∗}V = 0,

so that g is Einstein. To prove (2.25), (2.26) implies that V is a Killing field on (M, g) with V = 0 at ∂M and hence V = 0 on M, which gives (2.25).

The proof of the second statement is the same as above, by taking the linearization of Φ at g̃.

By Corollary 2.5, Einstein metrics in 𝔼_{B} satisfy the Bianchi gauge condition

(2.27)  βg̃(g) = 0,

on M. For C^{∞} metrics g, the proofs of Lemmas 2.3 and 2.4 give the splittings

(2.28)  S_{2}^{m,α}(M) = Kerδ ⊕ Imδ^{∗}, and S_{2}^{m,α}(M) = Kerβ ⊕ Imδ^{∗},

where δ = δ_{g} and δ^{∗} acts on χ_{1}^{m+1,α}, cf. [8]. However, the splittings (2.28) do not hold for a general metric g ∈ Met^{m,α}(M), since if Z ∈ χ_{1}^{m+1,α}, the derivative δ^{∗}Z = \frac{1}{2}L_{Z}g ∈ S_{2}^{m−1,α}(M), and not in S_{2}^{m,α}(M), so that there is a loss of one derivative. For Einstein metrics, this loss of derivative can be restored.

Lemma 2.6. For g ∈ 𝔼 = 𝔼^{m,α}(M), the splittings (2.28) hold. Moreover, given any g̃ ∈ 𝔼^{m,α} and g ∈ Met^{m,α}(M) nearby g̃, there exists a unique diffeomorphism ϕ ∈ D_{1}^{m+1,α}, close to the identity, such that

(2.29)  βg̃(ϕ^{∗}g) = 0.

In particular, ϕ^{∗}g ∈ Met_{B}^{m,α}(M).

Proof: For the first statement, by the proofs of Lemmas 2.3 and 2.4 and the discussion above, it suffices to prove that for any Z ∈ χ_{1}^{m+1,α}, L_{Z}g is C^{m,α} smooth up to ∂M, when g ∈ 𝔼^{m,α}. Einstein metrics are C^{∞} smooth in harmonic coordinates in the interior, and so L_{Z}g is C^{m,α} smooth in the interior of M. To show that L_{Z}g is C^{m,α} smooth up to ∂M, recall that in suitable boundary harmonic coordinates, one has

Δ_{g}g_{αβ} + Q_{αβ}(g, ∂g) = −2Ric_{αβ} = −2λg_{αβ},

cf. [4] for instance. Applying Z to this equation and commuting derivatives gives an equation for Δ_{g}Z(g_{αβ}) with 0 boundary values, (since Z(g) = 0 on ∂M), and with right-hand side in C^{m−2,α}. Elliptic boundary regularity results, cf. [9] for example, give Z(g_{αβ}) ∈ C^{m,α}, which implies that L_{Z}g is C^{m,α} smooth up to ∂M.

The second statement can be derived from the slice theorem of Ebin [8], but we give a direct and simpler argument here. (Note that the Lemma does not assert the existence of a slice per se). Let g̃ ∈ 𝔼^{m,α} and consider the map F : D_{1}^{m+1,α} × Met_{B}^{m,α}(M) → Met^{m,α}(M) given by F (ϕ, g) = ϕ^{∗}g. By the first part of the Lemma, F is C^{∞} smooth at g̃; F is linear in g and smooth in the direction of D_{1} at g̃ and hence smooth at g̃.

Now suppose g ∈ Met^{m,α}(M) is close to g̃. The linearization of F at (Id, g̃) is the map (X, h) → δ^{∗}X + h. As in the proof of Lemma 2.4, given any h, there exists a unique vector field Z ∈ χ_{1}^{m+1,α} such that on (M, g̃), β(δ^{∗}Z + h) = 0, or

βδ^{∗}Z = −β(h),

with respect to g̃. Hence, for g sufficiently close to g̃, there is a vector field Z such that

|βg̃(g + δ∗ ; g̃Z)| << βg̃(g).

It then follows from the implicit function theorem that there exists a diffeomorphism ϕ ∈ D_{1}^{m+1,α} such that (2.29) holds. It also follows that ϕ is unique, locally near the identity.

Next we return to the issue of ellipticity of the operator Φ in (2.17). The kernel of the linearized operator L in (2.18) forms the tangent space T_{g}Z_{B}, (g = g̃ here), and by Corollary 2.5,

(2.30)  T_{g}Z_{B} = T_{g}𝔼_{B},

so that the kernel also represents the space of (non-trivial) infinitesimal Einstein deformations in Bianchi gauge. The natural boundary conditions for L are

(2.31)  β(k) = 0, k^{T} = γ′ at ∂M.

However, perhaps surprisingly at first, the operator L with boundary conditions (2.31) does not form a well-defined elliptic boundary value problem. This is due to the well-known constraint equations, induced by the Gauss and Gauss-Codazzi equations on ∂M:

(2.32)  δ(A − Hγ) = −Ric(N, ·) = 0,
(2.33)  |A|^{2} − H^{2} + s_{γ} = s − 2Ric(N, N) = 0.

Here H = trA is the mean curvature and s and Ric are curvatures of the ambient metric g.

The tangential or vector constraint (2.32) will play a crucial rôle later on the rigidity issue itself. The normal or scalar constraint (2.33) is important in understanding the Fredholm properties of the boundary map Π in (2.6). Thus observe that for g ∈ 𝔼^{m,α}, one has A ∈ S_{2}^{m−1,α}(∂M) so that (2.33) implies that s_{γ} ∈ C^{m−1,α}(∂M). However, the space of metrics γ ∈ Met^{m,α}(∂M) for which s_{γ} ∈ C^{m−1,α}(∂M) is of infinite codimension in Met^{m,α}(∂M). This means that the linearization of the boundary map Π has infinite dimensional cokernel, at least when m < ∞, and so Π is not Fredholm. In particular it follows that the boundary conditions (2.31) are not elliptic.

Let

(2.34)  K = K_{g} = KerD_{g}Π.

Using the slice representation Z_{B} = 𝔼_{B} ⊂ 𝔼 at g̃ = g, K consists of forms κ such that

(2.35)  L(κ) = 0 and β_{g}(κ) = 0 on M, with κ^{T} = 0 on ∂M.

Again, since the operator L with boundary conditions (2.31) is not elliptic, it is not necessarily the case that K is finite dimensional.

Remark 2.7. It is useful to understand situations where the linearization DΠ has infinite dimensional kernel and cokernel, even in the C^{∞} case. Thus, consider the intersection K ∩ Imδ^{∗}. Let Y be a vector field at ∂M, (not necessarily tangent to ∂M), and extend Y to a vector field on M to be the unique solution to the equation β(δ^{∗}Y) = 0 with the given boundary value, cf. Lemma 2.4. Then L(δ^{∗}Y) = 0 and the boundary condition k^{T} = (δ^{∗}Y)^{T} = 0 is equivalent to the equation

(2.36)  (δ^{∗}Y^{T})^{T} + ⟨Y, N⟩A = 0 at ∂M.

If ∂M is planar, i.e. A = 0 on an open set U ⊂ ∂M, then this system has solutions of the form Y = fN, for any f with supp f ⊂ U, so that K ∩ Imδ^{∗} is infinite dimensional. Such vector fields Y thus are infinitesimal isometries at ∂M which do not extend to Killing fields (infinitesimal isometries) of (M, g). In particular, this shows that Theorem 1.3 is sharp, in that the hypothesis that ∂M is non-planar is necessary.

A similar phenomenon holds for the cokernel. Thus, suppose (∂M, γ) is planar with planar domain U ⊂ ∂M. Consider the linearization s′_{γ}(h), for h ∈ Im(DΠ). By differentiating the scalar constraint (2.33) in the direction h, one sees that s′_{γ}(h) = 0 on U, for any such h. This implies that ImDΠ has infinite codimension, even in the C^{∞} case, on such surfaces. The same argument and conclusion holds if A = 0 at just one point in ∂M.

To obtain an elliptic boundary value problem, we replace (2.31) with the boundary conditions

(2.37)  β(k) = 0, k_{0}^{T} = γ′_{0}, k_{00} = ω at ∂M,

where k_{0}^{T} is the trace-free part of k^{T} with respect to γ and k_{00} = k(N, N).

Proposition 2.8. The operator L with boundary conditions (2.37) is a well-defined elliptic boundary value problem of Fredholm index 0.

The proof is given in §3. There are other possible choices of elliptic boundary conditions, but none seems particularly preferable to the above. Given a background metric g̃ ∈ 𝔼^{m,α}, let S_{0}^{m,α}(∂M) be the space of C^{m,α} symmetric bilinear forms on ∂M which are trace-free with respect to the boundary metric γ̃. It follows from Proposition 2.8 that the map

(2.38)  Π̃ : 𝔼_{B} → S_{0}^{m,α}(∂M) × C^{m,α}(∂M),
Π̃(g) = (g_{0}^{T} , g(Ñ, Ñ)),

is Fredholm, of index 0, (for g near g̃). Let π_{1} be projection on the first factor in (2.38), and let Π_{0} = π_{1} ∘ Π̃. Thus Π_{0} = π_{0} ∘ Π, where π_{0} : S_{2}^{m,α}(∂M) → S_{0}^{m,α}(∂M) is the projection onto the trace-free forms with respect to g̃. In analogy to (2.34), let

(2.39)  K̃ = KerDΠ̃,

where the derivative is taken at g = g̃. In contrast to K in (2.34), K̃ is always finite dimensional. We will call an Einstein metric g ∈ 𝔼 (Dirichlet) non-degenerate if

(2.40)  K̃ = 0.

Thus, g is non-degenerate if and only if g is a regular point of the boundary map Π̃ in which case Π̃ is a local diffeomorphism near g.

Remark 2.9. It is useful to point out that the proof of Theorems 1.3 is quite simple when (M, g) is strongly non-degenerate, in the sense that K = 0 in (2.34). Thus, let X be a C^{m+1,α} vector field which is an infinitesimal isometry at (∂M, γ), so that (2.15) holds. (It is easy to see that this is well-defined, independent of any extension of X to M).

Choose any extension of X to a C^{m+1,α} vector field on M and let ϕ_{s} be a local curve of C^{m+1,α} diffeomorphisms of M̄ with ϕ_{0} = id such that \frac{d}{ds}ϕ_{s}|_{s=0} = X. The curve

g_{s} = ϕ_{s}^{∗}g

is a smooth curve in 𝔼 and by construction, [h] = [\frac{dg_{s}}{ds}] ∈ KerDΠ, for Π as in (2.6). One may then alter the diffeomorphisms ϕ_{s} by composition with diffeomorphisms ψ_{s} ∈ D_{1}^{m+1,α} if necessary, so that κ = \frac{dψ_{s}^{∗}(g_{s})}{ds} ∈ K_{g}, where K = K_{g} is the kernel in (2.34) and [h] = [κ]. Thus

κ = δ^{∗}X′,

where X′ = \frac{d(ϕ_{s}∘ψ_{s})}{ds} is C^{m+1,α} smooth up to M̄. Note that X′ = X at ∂M. If K_{g} = 0, then this gives

δ^{∗}X′ = 0 on M,

so that X′ is a Killing field on (M, g). Thus, any infinitesimal isometry at (∂M, γ) extends to a Killing field on (M, g), which completes the proof. It is worth pointing out that the condition (1.1) on π_{1} is not necessary in this situation.

The proof of Theorem 1.3 in general relies on an understanding of the cokernels of DΠ and DΠ̃. As a toy model, consider for example the operator

L(u) = Δu − λu,

on a bounded, smooth domain Ω ⊂ ℝ^{n} with Dirichlet boundary data. If H = L^{−1}(0), then one has a smooth map P : H → C^{m,α}(∂Ω), P (u) = u|_{∂Ω}. Suppose λ is an eigenvalue of the Laplacian and let K denote the λ-eigenspace, so that for k ∈ K, L(k) = 0 and k = 0 on ∂Ω. Then ImP is of codimension k = dimK in C^{m,α}(∂Ω). Observe that the functions

N(k)|_{∂Ω}, k ∈ K

span a slice for the cokernel of P; as above N is the outward unit normal at ∂M. There are several proofs of this; one proceeds as follows. (The proof of Proposition 2.10 below may also be adapted to give another proof). First, the Calderón unique continuation theorem [6], (for instance), implies that N(k) does not vanish identically on ∂Ω. Also, if N(k) denotes any extension of N(k)|_{∂Ω} to Ω, then the divergence theorem gives

∫_{Ω} L(N(k))k = ∫_{Ω} N(k)L(k) + ∫_{∂Ω} kNN(k) − (N(k))^{2}.

Since k ∈ K, L(k) = 0 and k = 0 on ∂Ω. Hence

∫_{Ω} L(N(k))k ≠ 0,

which implies that N(k) ∉ H, for any extension of N(k) on ∂Ω to Ω. Hence N(k)|_{∂Ω} ∈ CokerP, and the claim follows.

In the setting of Einstein metrics, the analogue of the quantity N(k) is (L_{N}k)^{T} at ∂M. Thus, define

(2.41)  C̃ = {((L_{N}κ)_{0}^{T} , N(κ_{00})) : κ ∈ K̃},

so that C̃ represents Neumann-type data associated with the Dirichlet data in (2.37) or (2.38).

Note that C̃ ⊂ S_{0}^{m,α}(∂M)×C^{m,α}(∂M). Namely, for κ ∈ K̃, one has L(κ) = 0 on M together with the elliptic boundary conditions β(κ) = 0, κ_{0}^{T} = 0, and κ_{00} = 0 on ∂M. Since g is C^{m,α} up to ∂M, elliptic boundary regularity applied to this system, (cf. [9], [11]), implies that L_{N}κ ∈ S_{2}^{m,α}(∂M) and N(κ_{00}) ∈ C^{m,α}(∂M).

Proposition 2.10. Given (M, g) with g ∈ 𝔼^{m,α}, the space C̃ is a slice for CokerDΠ̃ in S_{0}^{m,α}(∂M)× C^{m,α}(∂M), so that

(2.42)  S_{0}^{m,α}(∂M) × C^{m,α}(∂M) = ImDΠ̃ ⊕ C̃.

Proof: Let T = KerL, so that as in (2.30) and via Corollary 2.5, T is the space of infinitesimal Einstein deformations of g in Bianchi gauge on M. One has

(2.43)  T = K̃ ⊕ K̃^{⊥},

where K̃^{⊥} is the L^{2} orthogonal complement of K̃ in T. Let

P = DΠ̃ : T → S_{0}^{m,α}(∂M) × C^{m,α}(∂M)

be the modified boundary map: P (u) = (u_{0}^{T} , u_{00})|_{∂M}. Then P (K̃) = 0, while standard elliptic theory, (on compact manifolds with boundary), shows that P |_{K^{⊥}} : K̃^{⊥} → ImP is an isomorphism with bounded inverse. By Proposition 2.8, ImP = ImDΠ̃ is of codimension k = dimK̃ in S_{0}^{m,α}(∂M) × C^{m,α}(∂M).

Now suppose, for some 0 ≠ κ ∈ K̃, there exists u ∈ T such that

(2.44)  P (u) = ((L_{N})_{0}^{T} , N(κ_{00})).

The form u is determined only mod K̃, but one may choose u orthogonal to K̃. As above, let t(x) = dist_{g}(x, ∂M) and let κ^{T} be the restriction of κ to the level sets S(t) of t, (near ∂M). Then

(2.45)  (L_{N}κ)_{0}^{T} = − lim_{t→0} \frac{1}{t}κ_{0}^{T} , and N(κ_{00}) = − lim_{t→0} \frac{1}{t}κ_{00}.

More precisely, \frac{1}{t}κ_{0}^{T} is shorthand for \frac{1}{t}ϕ_{t}^{∗}(κ_{0}^{T}), where ϕ_{t} : ∂M → S(t) is the flow of ∇t = −N, and similarly for the second term in (2.45).

Let B(t) = {x ∈ M : t(x) ≥ t}, so that B(0) = M. Let T_{t} = T |_{B(t)}; as in (2.43), one has the splitting T_{t} = K ⊕ K_{t}^{⊥}, where K is restricted to B(t) and K_{t}^{⊥} is its orthogonal complement in T_{t}. On B(t) consider the form

u + \frac{1}{t}κ ∈ T_{t}.

By (2.44)-(2.45), one has ||P_{t}(u + \frac{1}{t}κ)||_{C^{0}(S(t))} → 0 as t → 0, where P_{t} denotes the corresponding boundary map at S(t). By the splitting (2.43) on T_{t}, there is a form ϕ_{t} ∈ K_{t}^{⊥} on B(t), with ||ϕ_{t}||_{B(t)} = o(1), such that

(2.46)  u + \frac{1}{t}κ = ϕ_{t} on B(t).

However, if κ ≠ 0, this is impossible, since the norm of u is bounded, the norm of ϕ_{t} is small as t → 0, while the norm of \frac{1}{t}κ is arbitrarily large. This contradiction proves the result.

We point out that exactly the same argument as above shows that any form ν = (L_{N}κ)^{T}, with κ ∈ K, (so that κ^{T} = 0 on ∂M), also satisfies ν ∉ ImDΠ. However, since Π is not Fredholm, (of index 0), it is not clear that such forms span the cokernel of DΠ.

By restricting to the first factor, it follows immediately from (2.42) that

(2.47)  S_{0}^{m,α}(∂M) = ImDΠ_{0} ⊕ Ñ_{0},

where Ñ_{0} = {(L_{N}κ)_{0}^{T} : κ ∈ K̃} and Π_{0} is defined following (2.38).

Next, we use the diffeomorphism group to pass from the space 𝔼_{B} of Bianchi-gauged Einstein metrics to the full space 𝔼, thus passing from Π̃ to the natural boundary map Π. Some preparatory remarks are useful in this respect. Thus, view S_{0}^{m,α}(∂M) ⊂ S_{2}^{m,α}(∂M) as the space of trace-free forms with respect to γ. If the boundary metric γ ∈ Met^{m+1,α}(∂M), then we claim that

(2.48)  S_{0}^{m,α}(∂M) + Imδ^{∗} + ⟨γ⟩ = S_{2}^{m,α}(∂M),

where δ^{∗} = (δ^{∗})^{T} acts on vector fields X ∈ χ^{m+1,α}(∂M) tangent to ∂M. First note that for γ ∈ Met^{m+1,α}(∂M), one has δ^{∗}X ∈ S_{2}^{m,α}(∂M). Tne proof of (2.48) then follows easily from the fact that the operator tr_{γ}δ^{∗} = −δ_{γ} is surjective onto functions of mean value 0 on ∂M, and thus has cokernel given by the constant functions. Thus, at least mod ⟨γ⟩, the extension of S_{0}^{m,α}(∂M) by infinitesimal diffeomorphisms of the boundary gives the full space S_{2}^{m,α}(∂M).

Of course the boundary map Π is equivariant with respect to diffeomorphisms, (in contrast to Π̃), and hence ImDΠ + Imδ^{∗} = ImDΠ. By combining (2.47) and (2.48), one might expect then that ImDΠ is always of finite codimension in S_{2}^{m,α}(∂M), (when γ ∈ Met^{m+1,α}(∂M)). However, Remark 2.7 shows that this is not the case; ImDΠ has infinite codimension whenever A = 0 at just some point of ∂M. The reason for this somewhat unusual behavior is that although Π_{0} = π_{0} ∘ Π, the map π_{0} is a projection map, and not an inclusion, and thus the setting of (2.47) is not quite the same as that of (2.48).

The next result establishes a weaker but still suitable form of this finite codimension statement, at least when the mean curvature H is non-vanishing on the boundary. However, instead of relying on diffeomorphisms tangent to ∂M as above, the proof rests crucially on the behavior of normal diffeomorphisms at ∂M. Because of this, one necessarily encounters the loss-of-derivative issue discussed following (2.28), with the consequence that one has to work with dense subspaces of S_{2}(∂M); this does not affect the main application of these results, which will occur in Lemma 2.14 below.

Thus, for the following result, we abuse notation slightly and consider deformations of the form h = δ^{∗}X where X = fN at ∂M to be in ImDΠ; hence DΠ is considered as a mapping into S_{2}^{m−1,α}(∂M) ⊃ S_{2}^{m,α}(∂M).

Proposition 2.11. Given (M, g) with g ∈ 𝔼^{m,α}, suppose that H is everywhere non-vanishing on ∂M. Then the inclusion map

(2.49)  ImDΠ + Ñ_{0} ↪ S_{2}^{m−1,α}(∂M)

has dense range. Moreover, there is a finite dimensional subspace K_{S} ⊂ K, with dimK_{S} = dimK̃, such that the image of the inclusion

(2.50)  ImDΠ + N_{S} ↪ S_{2}^{m−1,α}(∂M)

is also dense, where N_{S} = {(L_{N}κ)^{T} : κ ∈ K_{S}}.

Proof: Let T_{B} = ImDΠ(𝔼_{B}^{m,α}) ⊂ S_{2}^{m,α}(∂M). Then via (2.47), T_{B} projects onto S_{0}^{m,α}(∂M) modulo Ñ_{0}, and hence

(2.51)  S_{0}^{m,α}(∂M) ⊕ D = (T_{B} ⊕ Ñ_{0}) + D = S_{2}^{m,α}(∂M),

where D is the space of pure-trace forms fγ, f ∈ C^{m,α}(∂M).

Now consider the space F = {h ∈ S_{2}^{m−1,α}(∂M) : h = δ^{∗}(fN) = fA}. Extend fN to a vector field X on M by letting N near ∂M be the geodesic vector field normal to ∂M and requiring that N(fH) = 0 at ∂M, i.e. N(f) = −fN(H)/H, which is well-defined since H is non-vanishing. Note then that the inclusion

(2.52)  S_{0}^{m,α}(∂M) ⊕ F ⊂ S_{2}^{m−1,α}(∂M),

has dense range. This follows from the fact that for any h ∈ S_{2}^{m,α}(∂M), there exists a unique f, namely f = −trh/H, such that

(2.53)  ĥ = h + fA ∈ S_{0}^{m−1,α}(∂M).

Again, this is well-defined by the assumption that H ≠ 0 on ∂M.

Next observe that F is transverse to T_{B}. To see this, by the definition, it suffices to prove that if β(fA) = 0 on ∂M, then f = 0 on ∂M. Evaluating on N gives β(fA)(N) = δ(fA)(N)+ \frac{1}{2}N(fH) = δ(fA)(N), by the choice of extension above. Since A(N, ·) = 0, a simple computation shows that δ(fA)(N) = f|A|^{2}, which, via the hypothesis on H, vanishes on ∂M only if f vanishes on ∂M, as claimed.

Note that this transversality and (2.51)-(2.52) all also hold algebraically at each point of ∂M. It then follows that the linear projection of T_{B} along F into S_{0}^{m−1,α}(∂M) as in (2.53) maps onto a dense subspace of S_{0}^{m−1,α}(∂M) modulo Ñ_{0}, so that, analogous to (2.51), the inclusion

(T_{B} ⊕ Ñ_{0}) + F ⊂ S_{2}^{m−1,α},

has dense range. Since both T_{B} and F are contained in ImDΠ, this gives (2.49).

To prove (2.50), we first claim that pure-trace forms ϕγ on ∂M are either in ImDΠ or may be approximated arbitrarly closely by sequences in ImDΠ in the C^{m−1,α}(∂M) norm. To see this, for any such ϕ ∈ C^{m−1,α}(∂M), by (2.49) one may write

ϕγ ≈ k^{T} + ν_{0},

where k^{T} ∈ ImDΠ, ν_{0} ∈ Ñ_{0} and the approximation is in the C^{m−1,α}(∂M) topology. Since ν_{0} is trace-free, it follows that ϕ ≈ \frac{tr_{γ}k^{T}}{2} , so the equation above gives k_{0}^{T} ≈ −ν_{0}. But, by construction, k_{0}^{T} ∈ ImΠ_{0}, while ν_{0} ∉ ImΠ_{0} by (2.47). Hence, ν_{0} ≈ k_{0}^{T} ≈ 0 so that k^{T} ≈ ϕγ, which proves the claim.

Now suppose κ̃ ∈ K̃, so that in particular, κ_{0}^{T} = 0 on ∂M. Then by the preceding result, there exists k ∈ T 𝔼_{B} such that k^{T} ≈ −\frac{trκ^{T}}{2} γ at ∂M, with k ⊥ κ̃ on M, so that for κ = κ̃ + k, one has κ^{T} ≈ 0 on ∂M, i.e. κ is (arbitrarily) close to K. Taking the projection onto K gives an injective linear map

(2.54)  ι : K̃ → K, ι(κ̃) = κ.

Let K_{S} = ι(K̃). As noted following the proof of Proposition 2.10, any non-zero form ν ∈ N_{S} is not in ImDΠ, and since dimK_{S} = dimK̃, this gives (2.50).

Remark 2.12. For later purposes, we point out suitable local versions of Proposition 2.11 hold, when H vanishes somewhere on ∂M. Thus, let V ⊂ ∂M be a domain such that H ≠ 0 on V. Let S_{c}^{m−1,α}(V) ⊂ S^{m−1,α}(∂M) be the subspace of forms h of compact support in V. Then we claim there is a dense subspace W ⊂ S_{c}^{m−1,α}(V) such that, analogous to (2.50),

(2.55)  W ⊂ span(ImDΠ(V), N_{S}),

where ImDΠ(V) is the space of boundary values of infinitesimal Einstein deformations h with supp h ⊂ V. Since ImDΠ(V) = ImDΠ ∩ S_{c}^{m,α}(V), this follows immediately from (2.50) by taking the intersection with forms of compact support in V.

More generally, suppose V ⊂ ∂M is a subset on which A is non-vanishing. If H is non-vanishing on V, then we are in the case above, so suppose instead H = tr_{γ}A = 0 on V. In this case, there is a metric γ̂ (arbitrarily) close to γ such that Ĥ = trγ̂A ≠ 0 on any given subdomain V′ ⊂ V, and one may then apply exactly the same reasoning as above in the proof of Proposition 2.11 to prove (2.55) holds.

In sum Proposition 2.11 or more precisely (2.55) holds locally on domains in ∂M where the 2^{nd} fundamental form A is not identically 0.

Proposition 2.11 and Remark 2.12 provide the requisite (technical) information needed on the cokernel of DΠ. A key step in the proof of Theorem 1.3 is then the following formula or identity on (∂M, γ), which holds on any closed oriented Riemannian manifold. This result is proved in [1], [3], but for completeness, we give the full proof.

Proposition 2.13. Let X be a Killing field on (∂M, γ). Suppose τ is a divergence-free symmetric bilinear form on (∂M, γ). Then

(2.56)  ∫_{∂M}⟨L_{X}τ, h⟩dV_{γ} = −2 ∫_{∂M}⟨δ′τ, X⟩dV_{γ},

where L_{X} is the Lie derivative with respect to X and δ′ = \frac{d}{ds}δ_{γ+sh} is the variation of the divergence on (∂M, γ) in the direction h ∈ S_{2}(∂M).

Proof: Since the flow of X preserves γ, one has

(2.57)  ∫_{∂M}⟨L_{X}τ, h⟩dV_{γ} = − ∫_{∂M}⟨τ, L_{X}h⟩dV_{γ}.

Next, setting γ_{s} = γ + sh, the divergence theorem applied to the 1-form τ(X) on ∂M gives

(2.58)  0 = ∫_{∂M} δ_{γ_{s}}(τ(X))dV_{γ_{s}} = ∫_{∂M}⟨δ_{γ_{s}}τ, X⟩dV_{γ_{s}} − \frac{1}{2} ∫_{∂M}⟨τ, L_{X}γ_{s}⟩dV_{γ_{s}},

where the second equality is a simple computation from the definitions; the inner products are with respect to γ_{s}. Taking the derivative with respect to s at s = 0 and using the facts that X is a Killing field on ∂M and δτ = 0, it follows that

∫_{∂M}⟨δ′τ, X⟩dV − \frac{1}{2} ∫_{∂M}⟨τ, L_{X}h⟩dV = 0.

Combining this with (2.57) then gives (2.56).

A version of Proposition 2.13 also holds for infinitesimal isometries X which are not tangent to ∂M. For clarity in the exposition, we treat this case later, and so assume in the following, (until the Proof of Theorem 1.3 below), that X is tangent to ∂M.

At this point, the strategy is to examine the right side of (2.56) in connection with the divergence constraint (2.32); of course (2.32) implies the form τ = A − Hγ is divergence-free on ∂M. We first discuss the general perspective. Thus, one may view the pair (γ, A) as Cauchy data for the Einstein equations (2.3) at ∂M. The data (γ, A) are then formally freely specifiable subject to the constraints (2.32)-(2.33). For surfaces Σ ⊂ ℝ^{3}, the fundamental theorem of surfaces shows that the pair (γ, A) determines the embedding of Σ in ℝ^{3} up to congruence, i.e. isometry of ℝ^{3}, cf. [10], [17].

Now let T be the space of pairs (γ, τ) with τ divergence-free with respect to γ; here γ ∈ Met^{m,α}(∂M), τ ∈ S_{2}^{m−1,α}(∂M). The space T is naturally a vector bundle over Met^{m,α}(∂M),

(2.59)  π : T → Met^{m,α}(∂M),

with π the projection on the first factor. Let also F ⊂ T be the subset of pairs satisfying the scalar constraint equation (2.33). When expressed in terms of τ = A − Hγ, (2.33) is equivalent to

|τ|^{2} − (trτ)^{2} + s_{γ} = 0.

Pairs (γ, τ) ∈ F determine formal solutions of the Einstein, (or constant curvature), equations near ∂M. In more detail, let (t, x^{i}) be geodesic boundary coordinates for (M, g), so that by the Gauss Lemma, the metric g has the form

(2.60)  g = dt^{2} + g_{t},

where, as usual, t(x) = dist_{g}(x, ∂M) and g_{t} is the induced metric on the level set S(t) of t. Pulling back by the flow lines of ∇t, g_{t} may be viewed as a curve of metrics on ∂M, and one may formally expand g_{t} in its Taylor series:

(2.61)  g_{t} ∼ γ − tA − \frac{1}{2}t^{2}Ȧ + ⋯,

where Ȧ = ∇_{N}A = −∇_{T} A, T = ∇t = −N. As noted above, the terms (γ, A) are freely specifiable, subject to the constraint equations (2.32)-(2.33). All the higher order terms in the expansion (2.61) are then determined by γ and A; this can be easily seen by using the Riccati equation

(2.62)  ∇_{N}A + A^{2} + cg_{t} = 0,

and its successive derivatives along the t-geodesics; here c = \frac{λ}{2}. Thus, the formal series expansion for an Einstein metric g near ∂M is determined by the pair (γ, τ), τ = A − Hγ in F. The Cauchy-Kovalevsky theorem implies that if (γ, τ) are real-analytic forms on ∂M, then the formal series (2.61) converges to g_{t}, so that one obtains an actual constant curvature metric g as in (2.60), defined in a neighborhood of ∂M. Of course, such metrics will not in general extend to globally defined constant curvature metrics on M.

Now the right side of (2.56) is closely related to the linearization of the divergence constraint. Thus, if (γ_{s}, τ_{s}) is a curve in T with tangent vector (γ′, τ′) = (h, τ′) at s = 0, then one has

(2.63)  δ′(τ) + δ(τ′) = 0,

where δ′ is defined as in (2.56). This is the linearized divergence constraint.

Lemma 2.14. If the derivative Dπ in (2.59) is surjective at (γ, τ), τ = A − Hγ, then

(2.64)  L_{X}A = 0 on ∂M,

for any Killing field X on (∂M, γ). Moreover, (2.64) holds if ImDπ ⊂ S_{2}^{m,α}(∂M) is dense in S_{2}(∂M) with respect to the L^{2} norm. Conversely, if (2.64) holds for all such fields X, then Dπ is surjective.

Proof: The first statement follows easily from Proposition 2.13, with τ = A − Hγ. Thus, (2.63) gives δ′(A−Hγ) = −δ(τ′), for the variation δ′ of δ in any direction h ∈ T_{γ}Met(∂M). Hence, (2.56) gives

(2.65)  F(h) = ∫_{∂M}⟨L_{X}(A − Hγ), h⟩ = −2 ∫_{∂M}⟨δ(τ′), X⟩ = 2 ∫_{∂M}⟨τ′, δ^{∗}X⟩ = 0,

since X is a Killing field on (∂M, γ). Since h is arbitrary, this implies that

L_{X}(A − Hγ) = 0,

on ∂M, and (2.64) follows by taking the trace of this equation.

Next, F in (2.65) is a linear functional on S_{2}^{m,α}(∂M) ⊂ S_{2}(∂M), which is continuous in the L^{2} topology. Hence, if F(h) = 0 for h in a dense subspace of S_{2}(∂M) with respect to the L^{2} norm, then by continuity, F ≡ 0, which again gives (2.64).

The proof above also gives the converse as well, using the splitting (2.66) below.

Thus, given g ∈ 𝔼 and its corresponding 2^{nd} fundamental form A, giving the pair (γ, A) at ∂M, a basic issue is whether Dπ is surjective at (γ, A), i.e. whether the linearized divergence constraint (2.63) is solvable, for any variation h of γ on ∂M, (or for a space of variations dense in S_{2}(∂M) in the L^{2} norm). One cannot expect that this holds at a general pair (γ, τ) ∈ T . Namely, for any compact manifold ∂M, one has

(2.66)  Ω^{1}(∂M) = Imδ ⊕ Kerδ^{∗},

where Ω^{1} is the space of (C^{m−1,α}) 1-forms and δ^{∗} acts tangentially, as in (2.48). Thus, solvability at (γ, τ) in general requires that

(2.67)  δ′(τ) ∈ Imδ = (Kerδ^{∗})^{⊥}.

Of course Kerδ^{∗} is exactly the space of Killing fields on (∂M, γ), and so this space serves as a potential obstruction space.

Obviously, π is locally surjective when (∂M, γ) has no Killing fields. On the other hand, it is easy to construct examples where (∂M, γ) does have Killing fields and π is not locally surjective.

Example 2.15. Let (∂M, γ) be a flat metric on the 2-torus T^{2}; for example γ = dθ_{1}^{2} + dθ_{2}^{2}. Let τ = f(θ_{1})dθ_{2}^{2}. Then δτ = 0, for any C^{1} function f(θ_{1}). The pair (γ, τ) is in T , and in fact in F ⊂ T . Letting X be the Killing field ∂_{θ_{1}}, one has L_{X}τ ≠ 0 whenever f is non-constant, so that by the converse of Lemma 2.14, π is not locally surjective at such (γ, τ).

Returning to the situation of Einstein or constant curvature metrics, clearly Dπ is surjective onto ImDΠ, since ImDΠ consists of variations of the boundary metric determined by global variations of the Einstein metric g on M. Thus, one needs to examine the surjectivity of Dπ onto the cokernel CokerDΠ. Via Proposition 2.11, (cf. also Remark 2.12), one obtains:

Proposition 2.16. Suppose g ∈ 𝔼^{m,α} has boundary data (∂M, γ, A) which is not planar, i.e. A does not vanish identically on any open set U ⊂ ∂M. Then ImDπ ⊂ S_{2}^{m,α}(∂M) is dense in S_{2}(∂M) with respect to the L^{2} norm.

Proof: Suppose first that H is non-vanishing on ∂M, so that either H > 0 or H < 0 everywhere on ∂M. By Proposition 2.11, i.e. (2.50), it suffices to show that the linearized divergence constraint is solvable on N_{S}, since as remarked above, it is always solvable on ImDΠ. However, this follows essentially from the work already done in the proof of Propositions 2.10 and 2.11. Thus, first given any 0 ≠ κ ∈ K, (or K_{S}), consider the curve of metrics

(2.68)  g_{s} = g + s\frac{κ}{t},

for s small, on the annulus A(\frac{1}{2}t, 2t) = \{x ∈ M : \frac{1}{2}t ≤ t(x) ≤ 2t\}, where t > 0 is fixed for the moment. The curve g_{s} is Einstein, to 1^{st} order in s at s = 0. Hence, g_{s} satisfies the linearized divergence constraint (2.63) at s = 0, so that

(2.69)  δ′(A − Hγ_{t}) = −δ((A − Hγ_{t})′) on S(t),

where the derivatives are with respect to s, i.e. in the direction (\frac{κ}{t})^{T}; as above T denotes the restriction of a form to S(t). Here of course A = A_{t} and H = H_{t} are taken on S(t), as is δ. Recall also from (2.35) that κ^{T} = 0 on ∂M, so that κ^{T} = O(t) on S(t).

Now consider the limit of (2.68) and (2.69) as t → 0. One has γ_{t} → γ and A_{t}|_{S(t)} → A|_{∂M}. Also

(2.70)  lim_{t→0}(\frac{κ}{t})^{T} = −(L_{N}κ)^{T} .

Taking the limit of (2.69) at ∂M shows that the linearized divergence constraint is satisfied on N_{K} ⊃ N_{S}, as claimed. Note that the proof above that the linearized divergence constraint is solvable on N_{S} does not require that H is non-vanishing; it holds in general.

Next suppose (∂M, γ) is not planar. Then by Remark 2.12, and in particular (2.55), together with the preceding statement, the linearized divergence constraint is solvable on a dense set in S_{c}^{m−1,α}(V), where V equals the open dense set of ∂M on which A ≠ 0. However, S_{c}^{m−1,α}(V) is dense in L^{2} in S_{2}^{m−1,α}(∂M), which thus proves the result.

Combining the results above now easily proves Theorem 1.3.

Proof of Theorem 1.3: X tangent to ∂M.

By assumption, (∂M, γ) is not planar. Then Proposition 2.16 and Lemma 2.14 combine to prove that for any Killing field X at (∂M, γ), (i.e. infinitesimal isometry tangent to ∂M), one has

(2.71)  L_{X}A = 0,

as a form on ∂M. Now suppose X is C^{m+1,α} smooth on ∂M. By Lemma 2.4, X may be uniquely extended to a vector field X on M so that

(2.72)  βδ^{∗}X = 0 on M.

Since g ∈ 𝔼^{m,α}, the solution X is then C^{m+1,α} up to ∂M. Hence the form κ = δ^{∗}X is C^{m−1,α} up to ∂M and is an infinitesimal Einstein deformation in Bianchi gauge, i.e. L(κ) = L_{E}(κ) = 0 and β(κ) = 0. Note that by construction, κ ∈ K so that κ^{T} = 0 at ∂M. Calculating from the definitions, one easily sees that

(2.73)  (L_{X}A)^{T} = (A′_{δ^{∗}X})^{T} = (A′_{κ})^{T} ,

and so the equation (2.71) thus gives

(A′_{κ})^{T} = 0,

on ∂M. It then follows from Theorem 2.2 that κ = 0, and hence, δ^{∗}X = 0. Thus any Killing field on (∂M, γ) extends to a Killing field on (M, g). This proves Theorem 1.3, in the case X tangent to ∂M, assuming the validity of Theorem 2.2 and Proposition 2.8.

Proof of Theorem 1.3: General X.

The proof in the general case is essentially the same; in fact by means of a “thickening trick”, one can reduce the proof to the case where X is tangent to a boundary in one higher dimension.

Thus, suppose X is an infinitesimal isometry at ∂M and extend X to a vector field on M as in (2.72). As before, set κ = δ^{∗}X, so that κ^{T} = 0 on ∂M. Let ϕ_{t} be the flow of X, and for t small, set M_{t} = ϕ_{t}(M) ⊂ M′, ∂M_{t} = ϕ_{t}(∂M), with induced boundary metric γ_{t}. Here, since X is not tangent to ∂M, one needs to assume that the constant curvature manifold (M, g) embeds in a (slightly) larger constant curvature manifold (M′, g), and X is extended to M′. (This can of course always be accomplished).

For clarity, it is useful to first assume that ϕ_{t} is a curve of isometries on the boundary, so that ϕ_{t}^{∗}γ_{t} = γ, for t ∈ [−ε, ε] for some fixed ε > 0; thus, X_{t} = (\frac{d}{dt}ϕ_{t})_{t} is an infinitesimal isometry at the boundary ∂M_{t}. (This assumption, which is of course very strong in light of the fact that X is already fixed to satisfy (2.72), will be removed later).

Now consider the thickened manifold M̂ = I × M, I = [−ε, ε], and consider the metric

(2.74)  ĝ = dt^{2} + g_{t},

on M̂, where g_{t} = ϕ_{t}^{∗}g. We will work exclusively on the boundary ∂M̂ = I × ∂M, with induced boundary metric γ̂ = dt^{2} + γ_{t}. Let V be the vertical vector field on ∂M̂, V = ∂/∂t. Then

(2.75)  L_{V} γ̂ = 0 on ∂M̂,

i.e. V is a Killing field on (∂M̂, γ̂), exactly because X_{t} is an infinitesimal isometry at ∂M_{t} = {t} × ∂M.

Define the forms Â and τ̂ on ∂M̂ in the natural way. Thus, at {t}×∂M, Â is the 2^{nd} fundamental form of the boundary ∂M_{t} = ∂(ϕ_{t}(M)) in the metric g. In fact Â is the 2^{nd} fundamental form of ∂M̂ in M̂, since the t-direction of ∂M̂ is geodesic. Similarly, let τ̂ = Â − Hγ = Â − Ĥγ. Observe that τ̂ is divergence-free with respect to γ̂, i.e.

δ̂τ̂ = 0;

this follows from the divergence constraint (2.32), together with the fact that ( ˆ∇_{V} τ̂)(V, ·) = 0, (since V is tangent to geodesics and τ̂(V, ·) = 0). It follows that Proposition 2.13 holds for τ̂ on (∂M̂, γ̂), and (2.56) gives, for any form ĥ of compact support on ∂M̂, (i.e. ĥ = 0 at −ε × ∂M and ε × ∂M),

(2.76)  ∫_{∂M̂}⟨L_{V} τ̂, ĥ⟩dVγ̂ = −2 ∫_{∂M̂}⟨δ̂′τ̂, V ⟩dVγ̂.

Now restrict to infinitesimal deformations ĥ which preserve the splitting (2.74), so that ĥ is ’horizontal’ i.e. ĥ(V, ·) = 0. Then the variation δ̂′ ; ĥ = \frac{d}{ds}(δ_{γ̂+sĥ})_{s=0} on each \{t\} × ∂M is the same as the variation δ′_{h} = \frac{d}{ds}(δ_{γ_{t}+sh_{t}})_{s=0} on ∂M_{t}. Proposition 2.16 then applies and shows that for a dense set of such horizontal ĥ, (2.63) holds for each t, and hence δ̂′ ; ĥτ̂ = −δ̂(τ̂′), for some horizontal variation τ̂′. Proceeding then exactly as in the proof of Lemma 2.14 and using (2.75), it follows that

(2.77)  L_{V} Â = 0,

on ∂M̂. From the definitions above, this in turn implies that L_{X}A = 0 on ∂M, and the proof then follows from the same arguments as those following (2.71)

Finally, the proof above holds essentially unchanged if the assumption that X_{t} is an infinitesimal isometry on ∂M_{t} is dropped. In this case, the assumption that the initial vector field X is an infinitesimal isometry at (∂M, γ) implies that on ∂M̂ one has L_{V} γ̂ = O(t) on {t} × M in place of (2.75), so that (2.76) holds up to a term of order O(ε^{2}). Carrying out the arguments above for any ε > 0 and letting ε → 0 again gives (2.77).

This completes the proof of Theorem 1.3, (again modulo Theorem 2.2 and Proposition 2.8).

Proof of Theorems 1.1 and 1.2.

Consider first Theorem 1.1. The treatment of the cases 𝕊^{3}, ℝ^{3} and ℍ^{3} are identical and so we work in the setting of ℝ^{3}. Let F_{s}, s ∈ [0, 1], be a C^{1} curve of isometric embeddings of (Σ, γ) in ℝ^{3}, and let X_{s} = dF_{s}/ds be the corresponding curve of infinitesimal isometries along F_{s}(Σ). If the surfaces Σ_{s} = F_{s}(Σ) ⊂ ℝ^{3} are not planar for any s, (or for a set of s of full measure), then Theorem 1.1 follows immediately from Theorem 1.3 by integration in s, since Theorem 1.3 implies that X_{s} is then the restriction of a Killing field on ℝ^{3} to Σ_{s}. Moreover, the set of s ∈ [0, 1] for which X_{s} is the restriction of a Killing field on ℝ^{3} to Σ_{s} is clearly closed in [0, 1].

Thus, (without loss of generality), we may suppose that each surface Σ_{s} is planar in ℝ^{3}. Let Z_{s} = {x ∈ Σ_{s} : A_{s}(x) = 0} be the planar region in Σ_{s}. These planar regions are mapped smoothly into themselves along the curve of isometric embeddings, and hence

(2.78)  L_{X_{s}}A_{s} = 0,

on Z_{s}. On the other hand, as discussed in the proof of Proposition 2.16 in the paragraph following (2.70), the linearized divergence constraint is solvable on a dense subset of S_{c}^{m−1,α}(V_{s}), where V_{s} = Σ_{s} \ Z_{s}. It then follows from (the proof of) Lemma 2.14 that (2.78) holds also on V_{s}, and hence it holds everywhere on Σ_{s}. Thus, again as in the proof of Theorem 1.3 and via Theorem 2.2, X_{s} extends to a Killing field of ℝ^{3} for any s and the result follows again by integration in s.

The proof of Theorem 1.2 is essentially the same, again via Theorem 2.2.

3. The Banach manifold 𝔼 and Degeneracy

In this section, we prove Theorems 2.1 and 2.2. It turns out that these results, and their proofs, are closely related. As mentioned above, overall the strategy to prove Theorem 2.1 is to use the implicit function theorem. To do this, we first need to prove Proposition 2.8, which is stated in equivalent form below. As before, throughout this section, all linearizations are taken at g̃ = g.

Proposition 3.1. For any g near g̃, the operator Φ = Φg̃ in (2.17) with boundary conditions

(3.1)  βg̃(g) = 0, (g|_{T (∂M)})_{0} = γ_{0}, g(Ñ, Ñ) = ω on ∂M.

forms a well-defined elliptic boundary value problem.

Proof: It suffices to compute the symbol of the linearization of the equations at g = g̃ and at the flat metric. The leading order symbol σ(L′) is then the same as the full symbol σ(L), for L as in (2.18). The symbol σ(L) is a 6 × 6 matrix, acting on a column vector h_{αβ}. The rows and columns are ordered lexicographically (αβ), as (00), (0i), (ij), 1 ≤ i ≤ j ≤ 2. Then

(3.2)  σ(L) = −|ξ|^{2}I,

where I is the 6 × 6 identity matrix.

The symbol σ(B′) for the leading order part of the boundary operator B has the following form. The linearization of the first term in (3.1) is β(h) = δh + \frac{1}{2}dtrh, with symbol acting on h given by −zh_{0α} − ξ_{j}h_{jα} + \frac{1}{2}ξ_{α}h_{ββ}, where z = ξ_{0}. This represents the first 3 rows of σ(B′), with 0 ≤ α ≤ 2. The linearization of the second term in (3.1) is the trace-free part of h^{T}, (with respect to γ̃), and so given by h_{ij} − \frac{1}{2}h_{kk}δ_{ij}. Finally the symbol for the third term in (3.1) is 1 in the (22) row, and (00) column. This gives the matrix:

(3.3)  σ(B′) = ( \begin{matrix}−\frac{1}{2}z & −ξ_{1} & −ξ_{2} & \frac{1}{2}z & 0 & \frac{1}{2}z \\ \frac{1}{2}ξ_{1} & −z & 0 & −\frac{1}{2}ξ_{1} & −ξ_{2} & \frac{1}{2}ξ_{1} \\ \frac{1}{2}ξ_{2} & 0 & −z & \frac{1}{2}ξ_{2} & −ξ_{1} & −\frac{1}{2}ξ_{2} \\ 0 & 0 & 0 & \frac{1}{2} & 0 & −\frac{1}{2} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & −\frac{1}{2} & 0 & \frac{1}{2} \end{matrix} )

To check that this forms an elliptic boundary system, following [11], set z = ξ_{0} in σ(L) and for ξ = {ξ_{i}} ∈ T^{∗}(∂M), let L_{0}^{+}(z, ξ) be the polynomial formed from the positive imaginary roots of the interior operator L, so that

L_{0}^{+}(z, ξ) = (z − i|ξ|)^{6}.

Let L^{∗} be the matrix adjoint of σ(L), so that L^{∗} = detL · L^{−1}. (Here and below we abuse notation and write L for σ(L), and B for B′, etc.). One then forms the matrix of polynomials

Q(z, ξ) = B · L^{∗};

the entries of Q are polynomials in z, for any fixed ξ ∈ T_{x}^{∗}(∂M). Ellipticity then requires that the rows of Q = {Q_{μν}} are linearly independent polynomials modulo the polynomial L_{0}^{+}(z, ξ), i.e. no non-trivial linear combination of the rows of Q is a 6-tuple with a common factor of L_{0}^{+}. Since L is diagonal, this is equivalent to the statement that the rows of B are linearly independent modulo (z − i|ξ|), i.e. detB ≠ 0 modulo (z − i|ξ|), which is easily verified from the form of B above in (3.3).

A similar calculation shows that the boundary data (3.1) may be continuously deformed to full Dirichlet boundary data g_{αβ} = γ_{αβ} maintaining ellipticity. The latter boundary value problem clearly has index 0, and hence, by the homotopy invariance of the index, so does the boundary system (3.1).

Next, we show that L surjects onto S^{2}(M) when its domain is expanded from Met_{B}(M) to all of Met(M).

Lemma 3.2. At any g̃ ∈ 𝔼^{m,α}, the linearized operator L = 2DΦ:

(3.4)  L : Tg̃Met^{m,α}(M) → S_{2}^{m−2,α}(M)

is surjective.

Proof: Since L is elliptic, and so Fredholm, if L is not surjective, then there exists a non-zero k ∈ Tg̃Met(M) = S_{2}(M) such that, for all h ∈ S_{2}(M),

(3.5)  ∫_{M}⟨L(h), k⟩ = 0.

Since L is formally self-adjoint, this gives

(3.6)  0 = ∫_{M}⟨L(h), k⟩ = ∫_{M}⟨h, L(k)⟩ + ∫_{∂M} D(h, k),

where

(3.7)  D(h, k) = ⟨h, ∇_{N}k⟩ − ⟨k, ∇_{N}h⟩.

The form h is arbitrary, and hence both the bulk integral and the boundary integral on the right in (3.6) vanish separately. Thus

(3.8)  L(k) = D^{∗}Dk − 2R(k) = 0,

and since h and ∇_{N}h are arbitrary at ∂M,

(3.9)  k = ∇_{N}k = 0 at ∂M.

Thus the form k solves the elliptic equation (3.8) and has vanishing Cauchy data at ∂M. It then follows from the Calderón unique continuation theorem, cf. [6], that k = 0 in a neighborhood U of ∂M in M. Since the operator L has analytic coefficients, (in harmonic coordinates), standard analytic continuation in the interior implies that k = 0 on M, which proves the result.

Proposition 3.3. Suppose either π_{1}(M, ∂M) = 0 or M ⊂ M_{c}^{3} with ∂M connected. Then at any g̃ ∈ 𝔼^{m,α}, the map Φ = Φg̃ : Met_{B}^{m,α}(M) → S_{2}^{m−2,α}(M) is a submersion at g̃, i.e. the derivative

(3.10)  L : Tg̃Met_{B}^{m,α}M → S_{2}^{m−2,α}(M)

is surjective, and its kernel splits.

Proof: To see that L in (3.10) is surjective at g̃ = g, as in the proof of Lemma 3.2, if k ⊥ ImL|_{T Met_{B}(M)}, then

(3.11)  L(k) = 0.

We claim that the boundary conditions analogous to (3.9) are now weakened to

(3.12)  k^{T} = 0, (A′_{k})^{T} = 0.

This can be verified by a direct, although lengthy, computation, given in [2]. More conceptually, by (3.9), D(h, k) = 0, for all h ∈ Met(M), if and only if k = O(t^{2}) near ∂M. Via Lemma 2.6, write h ∈ T_{g}Met(M) as h = h_{B} + δ^{∗}X, where h_{B} ∈ T_{g}Met_{B}(M). The vanishing of D(h, k) on T_{g}Met_{B}(M) is then equivalent to the vanishing of the gauge-invariant part of D(h, k), which implies that k = δ^{∗}Z + O(t^{2}) near ∂M. By (2.13), this is equivalent to (3.12).

Next we prove that k is an infinitesimal Einstein deformation, which follows from the claim

(3.13)  β(k) = 0.

First, we show that

(3.14)  δk = 0.

To see this, note that for any vector field X on M, L(δ^{∗}X) = δ^{∗}Y, where Y = 2βδ^{∗}X. This follows from the relation (2.19) between L and L_{E}, together with the fact that L_{E}(δ^{∗}X) = 0. Thus (3.5) gives

0 = ∫_{M}⟨L(δ^{∗}X), k⟩ = ∫_{M}⟨δ^{∗}Y, k⟩.

The condition δ^{∗}X ∈ T Met_{B}(M) is equivalent to Y = 0 at ∂M. As in the proof of Lemma 2.3, the operator δδ^{∗} is elliptic, with trivial kernel on χ_{1}, and hence surjective. Thus, Y above may be any vector field on M. Applying the divergence theorem to the equation above then gives (3.14).

Next regarding trk, the constraint β(h) = 0 on h, written out in tangential and normal components, has the form

(3.15)  (∇_{N}h)(N)^{T} = δ^{T} h^{T} − α(h(N)) + \frac{1}{2}d^{T} trh,
(3.16)  N(h_{00}) = δ^{T} (h(N)^{T}) − h_{00}H + ⟨A, h⟩ + \frac{1}{2}N(trh),

where α(h(N)) = A(h(N)) + Hh(N)^{T}, and N(h_{00}) = (∇_{N}h)(N, N).

First, one may choose h = 0 on ∂M and ∇_{N}h to satisfy

(∇_{N}h)(N)^{T} = 0 and N(h_{00}) = ⟨∇_{N}h, γ⟩,

where we have used the fact that trh = h_{00} + tr_{γ}h. The constraints are then satisfied and by (3.6), one has

∫_{∂M}⟨∇_{N}h, k⟩ = 0,

for all such h. Using (3.12), this gives

(3.17)  k_{00} = trk = 0,

on ∂M. Next taking the γ-trace of the second equation in (3.12) and using (3.17) gives

⟨∇_{N}k, γ⟩ = −2δ^{T} (k(N)^{T}).

On the other hand, expanding (3.14) and using the first equation in (3.12) gives

N(k_{00}) = δ^{T} (k(N)^{T}),

so that

⟨∇_{N}k, γ⟩ = −2N(k_{00}).

Returning now to the constraints (3.15)-(3.16), set h^{T} = fγ, N(h_{00}) = fH, with the other components of h and ∇_{N}h equal to 0. This also satisfies the constraints, for any f and it follows from the equation D(h, k) = 0 and (3.17) that

⟨∇_{N}k, γ⟩ = 0.

Combining the last two equations gives then

(3.18)  N(trk) = 0 at ∂M.

Finally, taking the trace of (3.11) on M gives

Δtrk + \frac{s}{n + 1}trk = 0.

Since by (3.17) and (3.18) the Cauchy data for trk vanishes on ∂M, it follows from the Calderón unique continuation theorem that trk = 0 on M, so that (3.13) holds.

So one needs to show that the conditions (3.11)-(3.13) imply that k = 0, under either of the conditions π_{1}(M, ∂M) = 0 or M ⊂ M_{c}^{3} with ∂M connected. Note that this is just the statement of Theorem 2.2. By (3.12) and (3.14), k is an infinitesimal Einstein deformation L_{E}(k) = 0. In dimension 3, Einstein metrics are of constant curvature, and its well-known that any infinitesimal constant curvature deformation is locally pure gauge, cf. [17] for instance. Thus, locally everywhere on M, and globally if M ⊂ M_{c}^{3}, one has

(3.19)  k = δ^{∗}Y,

for some (local) vector field Y. The vector field Y is unique modulo local Killing fields on (M, g). Of course space-forms always have local Killing fields. The equation L(k) = 0, or equivalently βδ^{∗}Y = 0, has analytic coefficients in g-harmonic coordinates, (since g is analytic in harmonic coordinates), and hence k or Y is analytic in such coordinates. If

(3.20)  π_{1}(∂M) → π_{1}(M) → 0,

then standard analytic continuation shows that Y in (3.19) is uniquely determined by its values on the component ∂_{0}M of ∂M containing the base point for π_{1}(∂M). On the other hand, this statement holds automatically, (without the assumption (3.20)), if M ⊂ M_{c}^{3}.

In either case, we then claim that the boundary conditions (3.12) imply that there is a unique vector field Y on M, with Y = 0 on the given component ∂_{0}M of ∂M, satisfying (3.19) globally on M. To see this, first pass to geodesic gauge. Thus, given k, define Y near ∂M by

(3.21)  [N, Y] = δ^{∗}Y (N, ·) = k(N),

with Y = 0 on ∂M. The equation (3.21) is an ODE for Y along the geodesics tangent to N and so has a unique solution Y with Y = 0 on the given component of ∂M. Set k̂ = k − δ^{∗}Y. Then k̂ is an infinitesimal Einstein deformation with

(3.22)  k̂(N, ·) = 0,

i.e. k̂ is in geodesic gauge near ∂M. In this gauge, the boundary conditions (3.12) give

(3.23)  k̂ = 0, ∇_{N} k̂ = A′ ; k̂ = 0, at ∂M.

Along the geodesics normal to ∂M, one has the Riccati equation (2.62). Expressed in terms of the Lie derivative L_{N}A, this is

(3.24)  L_{N}A − A^{2} + cg_{t} = 0.

Taking the variation in the direction k̂, one has (L_{N}A)′ = L_{N}A′ and (L_{N}L_{N}g)′ = L_{N}L_{N} k̂. Hence, the variation of (3.24) in the direction k̂ has the form

(3.25)  \frac{1}{2}L_{N}L_{N} k̂ − A ∘ L_{N} k̂ + ck̂ = 0.

This is a 2^{nd} order linear ODE along the geodesics normal to ∂M, with vanishing Cauchy data (3.23) at ∂M. Thus one has k̂ = 0, i.e.

(3.26)  k = δ^{∗}Y,

near ∂M, with Y = 0 at ∂M. Via the previous discussion concerning (3.19) and (3.20), this proves the claim above.

It follows that if, in addition to (3.20), ∂M is connected, i.e. π_{1}(M, ∂M) = 0, or ∂M ⊂ M ⊂ M_{c}^{3} is connected, then (3.26) holds globally on M, with Y = 0 on all of ∂M. Now the same elementary integration-by-parts argument as in the proof of Lemma 2.3 or 2.4 then proves Y = 0 on M and hence k = 0. This proves the surjectivity statement.

It is essentially standard or formal that the kernel of DΦ splits, i.e. it admits a closed complement in T_{g}Met_{B}(M), cf. [2] for the proof in detail.

As noted above, the proof of Proposition 3.3 also proves Theorem 2.2. (It is easily verified that the proof above also holds for k ∈ S_{2}^{2}(M)). It is now simple to complete the proof of Theorem 2.1.

Proof of Theorem 2.1.

Proposition 3.3 together with the implicit function theorem in Banach spaces shows that the local spaces Z_{B} = 𝔼_{B} are smooth Banach manifolds. By Lemma 2.6, 𝔼_{B} is thus a local smooth slice to the action of D_{1} on 𝔼. These local spaces give local coordinate patches for E. The overlap maps are smooth, since the gauge condition βg̃(g) = 0 varies C^{∞} smoothly with g̃. This shows that E is a C^{∞} smooth Banach manifold, (and Fréchet manifold when m = ∞). It is clear that the boundary map Π in (2.6) is C^{∞} smooth.

Remark 3.4. It remains an open question whether the space E is a smooth Banach manifold when the condition π_{1}(M, ∂M) = 0 is dropped. However, without this condition, there are simple examples which show that Φ is not a submersion. Thus, for instance, let M = S^{3} \ (B_{1} ∪ B_{2}), where B_{i} are a pair of disjoint 3-balls in S^{3}. By the Weyl embedding theorem in this context, cf. [12], [13], the space E is a smooth Banach manifold in a neighborhood of a round metric on S^{3}, and round metrics on ∂M. However, using the isometry group of S^{3}, one may move one 3-ball B_{2} with respect to the other B_{1} in S^{3}, giving a 2-dimensional family of isometrically distinct metrics on which the boundary map Π is constant. The corresponding 2-dimensional space of infinitesimal variations gives Einstein deformations κ satisfying (3.12), but which are not pure D_{1}-gauge globally, i.e. (3.26) does not hold with Y = 0 on ∂M. Thus, DΦ has a non-trivial cokernel; Theorem 2.2 fails without this condition.

Similarly, DΠ is surjective in this example, but has (at least) a 2-dimensional kernel, so that the Fredholm index of DΠ is (at least) 2.

Remark 3.5. The attentive reader will have noticed that much of the work above holds for Einstein metrics in any dimension n + 1 ≥ 3. This is of course the reason why most of the results have been formulated in the context of Einstein, as opposed to constant curvature, metrics. In fact, the only the statement and proof above which does not essentially immediately hold for general Einstein metrics is the local gauge statement (3.19). The validity of Theorems 1.1 and 1.2 in this situation will be discussed elsewhere.

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July/October, 2007

Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 E-mail: anderson@math.sunysb.edu