We will see in Section 4.3.3 —see Proposition 8— that non-existence of tracefree Codazzi tensors is, in a sense, stable under perturbations of the metric. This will be used in the sufficiency argument of the same section —see Proposition 9.
Proposition 2. Let h be a Riemannian metric of non-negative sectional curvature. If ηij is a Codazzi tensor of constant trace, then it is covariantly-constant —i.e. Dkηij = 0. Moreover, if the sectional curvatures are not vanishing everywhere, then ηij is a constant multiple of hij.
The proof follows by establishing positivity of the integrand in the following identity ∫ S ( ∥DY ∥2+ r ijYkiYkj− rikjlYijYkl ) dµ (4.1.17)
—see [42] for full details.
4.2 Conformally-rigid hyperbolic initial data
From the previous two sections, we know that the existence of either a non-trivial conformal Killing vector or a non-trivial tracefree Codazzi tensor is undesirable for the application of the Friedrich– Butscher method on compact manifolds. Moreover, it was noted earlier that a Riemannian manifold of negative-definite Ricci curvature cannot admit a globally-defined conformal Killing field, rendering such a manifold a natural first candidate for the background manifold (S,˚h).
Due to the highly-coupled nature of the auxiliary system of equations, Ψ = 0, the tractability of the required analysis is, of course, dependent on the specific properties of the background manifold, (S,˚h). In particular, if we consider a manifold (S,˚h) that is Einstein (or, equivalently, a space form
since we are in dimension 3):
˚rij =13˚r˚hij,
with ˚r (necessarily) constant, then DvΨ simplifies significantly. The requirement that ˚e rijbe negative- definite is then simply that ˚r be negative.
Moreover, we would also like to exclude the possibility of a non-trivial tracefree Codazzi tensor —i.e. to ensure that ker( ˚D) ∩ S2
0(S;˚h) ={0}. Now, in the case of hyperbolic manifolds —see [57]
and also also [58]— the space of tracefree Codazzi tensors coincides with the space of essential
conformally flat deformations —i.e. one has
ker{˚D : S2
0(S;˚h)→ J (S)} ≃ ker H/˚L(Λ 1(S)),
where H denotes the linearised Cotton map —see Section 4.4 for more details. Hence, in the search for a suitable background metric, we are naturally led to the notion of conformal rigidity —see Definition 11— the requirement of which places additional restrictions on the topology ofS.
4.2.1
Closed hyperbolic 3-manifolds
Hyperbolic manifolds remain an active area of research in geometry and topology, due to their importance in the decomposition of 3−manifolds according to the Geometrisation Conjecture.
Definition 10. A hyperbolic 3−manifold, (S, h), is a manifold of dimension 3 equipped with a
hyperbolic Riemannian metric —i.e. an Einstein metric of (constant) negative scalar curvature. 3
3In dimensions n > 3, the term hyperbolic is reserved for manifold of constant negative sectional curvature, a
In this chapter, we will be concerned with closed hyperbolic 3−manifolds. We recall here for
context some relevant theorems of Riemannian geometry, the proofs which are far beyond the scope of this thesis —we refer the interested reader to [59] for more details. Recall that the Killing–Hopf Theorem implies that a complete, connected hyperbolic manifold is necessarily isometric to a quotient of the hyperbolic space,Hn, by a discrete group of its isometries —i.e. a Kleinian group. Moreover, in the case of compact manifolds, completeness is guaranteed automatically by the Hopf–Rinow Theorem, and hence the manifolds of interest here possess a metric which is locally of the following form
ds2= 4(dx
2+ dy2+ dz2)
(1− (x2+ y2+ z2))2.
—i.e. locally isometric to the hyperbolic plane. While the requirement of positive curvature imposes strict topological restrictions on S —see [42]— the requirement of negative curvature is much less
restrictive. One procedure for the construction of hyperbolic 3−manifolds is that of Dehn surgery:
one removes a link, L , from S3 and then glues in 2−tori (one for each connected component of
the link) by identification of their boundaries with ∂(S3\ L ) —the ways in which the boundaries
may be identified are parametrised by a set of slopes, one for each connected component of the link. A theorem of Lickorish–Wallace (see [59]) guarantees that every 3−manifold topology may
be obtained via Dehn surgery on some link in S3. A theorem of Thurston (see Section 6.26 [42])
establishes then that, ifS3\ L admits a complete hyperbolic metric, so too do the Dehn-surgered
manifolds, for all but finitely many choices of slopes. Moreover, ifL is a knot, another theorem of
Thurston establishes that the manifoldS3\ L admits a complete hyperbolic metric if and only if L
is not a torus knot or a satellite knot. When it exists, the hyperbolic metric on the Dehn-surgered (or Dehn-filled) manifold is unique by virtue of the Mostow Rigidity Theorem.
For what follows, we will be interested in the manifolds resulting from Dehn surgery on knots since the existence of Codazzi tensors on such manifolds has been addressed in [60] —see the next section.
Remark 14. For the purposes of Chapter 7, we note here that the manifolds resulting from Dehn
surgery on a knot have vanishing first Betti number (b1 = 0). Intuitively, this can be understood
from the fact that all holes are filled when the 2−torus is glued in; the resulting cohomology contains
only torsion groups, arising from the twisting of the 2−torus boundary, as determined by the slopes
prescribed in the gluing procedure. This may be demonstrated more rigorously using the Mayer–
Vietoris sequence. The fact that b1= 0 will be important when we study the full CCEs in Chapter
7 since, by Hodge’s Theorem (see [61], for example), the Betti number is equal to the nullity of the Hodge Laplacian on 1−forms, which we will need to be an isomorphism for application of the IFT.
4.2.2
Conformal rigidity and the (non-)existence of Codazzi tensors
Following from the discussion in Section 4.1.4, a closed hyperbolic manifold does not admit any global conformal Killing vector fields, since the Ricci tensor in this case is negative-definite. It remains to investigate the existence of tracefree Codazzi tensors; we will see that the question of the existence of tracefree Codazzi tensors on a hyperbolic manifold is connected to its conformal properties under metric perturbations. More precisely, the space of tracefree Codazzi tensors on a hyperbolic manifold is precisely the space of essential conformally flat deformations —by essential, we mean L2−orthogonal to metric perturbations of the form ˚L(X) (see Remark 15).
4.2. Conformally-rigid hyperbolic initial data 61
Consider first the linearisation of the Cotton–York tensor,H[h]ij, about a background metric ˚h:
H(η)ij≡ d dτH(˚h + τ η)ij τ =0 = ˚ϵkl(i( ˚D|kDRic(η)l|j)− C(η)m|k|j)˚rlm) + η(ikH˚j)k−12η ˚Hij with indices raised using ˚hij. Here, η ≡ tr
˚h(η), the operator C(·) i
jk is the linearisation of the Christoffel symbols, as in (4.1.7), and DRic(η)ijis the linearised Ricci operator acting on the metric perturbation ηij —see equation (4.1.6).
Definition 11. Following [62], a conformally flat manifold (S, h) will be said to be conformally-rigid
if the space of essential conformally flat deformations is trivial —i.e. if ker H/˚L(Λ1(S)) = {0},
where H denotes the linearised Cotton map —see Section 4.4.
Remark 15. Note that, indeed, Im ˚L⊆ ker H, as a consequence of the fact that the Cotton–York
tensor is conformally-covariant and a metric perturbation in Im ˚L corresponds to an infinitesi-
mal conformal diffeomorphism —see Section 4.4.1 for more details. The space ker H/˚L(Λ1(S)) is
sometimes called the premoduli space of conformally-flat structures around [˚h]. By the “Splitting
Lemma”, Lemma 3,
ker H/˚L(Λ1(S)) ≃ ker H ∩ ker ˚δ.
The connection between the notion of conformal-rigidity and the (non-)existence of tracefree Codazzi tensors is made precise by the following Proposition from [58], the proof of which given in Appendix A.1.
Proposition 3. Let (S,˚h) be a closed hyperbolic manifold, then
ker{˚D : S2
0(S;˚h)→ J (S)} = ker H ∩ ker ˚δ.
Hence, if ˚h is conformally-rigid then it admits no non-trivial tracefree Codazzi tensors.
The existence of a family of such manifolds is guaranteed by a theorem of Kapovich —see [60]— which states, roughly, that for all but finitely-many choices of slope, s, the Dehn-filled hyperbolic manifoldS = M(s) is conformally-rigid. The proof is beyond the scope of this thesis.
The results of the previous section can be summarised as follows:
Proposition 4. Let (S,˚h) be a closed hyperbolic, conformally rigid manifold. Then (S,˚h) admits
neither global conformal Killing vectors nor global tracefree Codazzi tensors.
4.2.3
The background initial data sets
In the following, let (S,˚h) be a closed hyperbolic manifold with sectional curvature normalised to
k =−1 (or, equivalently, with ˚r =−6). Then, for any given constant ˚K, the tensor fields
˚hij, K˚ij= 1
overS constitute a solution to the Einstein constraint equations with constant mean extrinsic cur-
vature ˚K and with cosmological constant given by λ = 13( ˚K2− 9),
as can be readily seen from the Hamiltonian constraint (1.2.1a). Initial data of this type will be called hyperbolic initial data. We remark in passing that it was shown in [62] that the subclass of such initial data with λ = 0 is Cauchy stable in the expanding time-direction.
Remark 16. Note that here we are choosing to normalise the intrinsic curvature, which in turn
fixes the value of the cosmological constant, once the extrinsic curvature has been given. One could alternatively rescale the intrinsic and extrinsic curvatures appropriately so as to normalise the cosmological constant. The former option is chosen since, in the subsequent analysis, it is the intrinsic geometry of (S,˚h) that will be of primary importance.