Local gauge conditions for ellipticity in conformal geometry

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.


Introduction
In Riemannian geometry, various regularity results for curvature tensors may be obtained via harmonic coordinate systems.We recall the basic idea following [7], [10], [26].
Let (M, g) be an n-dimensional Riemannian manifold, and let (x 1 , . . ., x n ) be a local coordinate system.The Ricci tensor of g has the expression where ∆ is the Laplace-Beltrami operator, Γ l = g ab Γ l ab , and Γ l = g la Γ a .One also has Γ l = −∆x l .If the coordinate system is harmonic, in the sense that ∆x l = 0 for all l, then the Ricci tensor becomes an elliptic operator.Thus by elliptic regularity, if the Ricci tensor is smooth in harmonic coordinates, then also the metric is smooth in these coordinates.In particular, if the Riemann curvature tensor of a low regularity metric vanishes, this implies that the metric is smooth in harmonic coordinates and thus locally flat by classical arguments.
The Ricci tensor is of course invariant under diffeomorphisms, and these diffeomorphisms correspond to different gauges for the equation Ric(g) = h.The choice of harmonic coordinates may be viewed as a local gauge condition that results in an elliptic equation.In this article we give similar local gauge conditions for the ellipticity of conformal curvature tensors, based on systems of n-harmonic coordinates.We say that a function u in (M, g) is p-harmonic (1 < p < ∞) if δ(|du| p−2 du) = 0.
Here | • | is the g-norm and δ is the codifferential.A local coordinate system is called p-harmonic if each coordinate function is p-harmonic (the case p = 2 of course corresponds to the usual harmonic coordinates).
In the earlier article [21], we established the existence of p-harmonic coordinate systems for 1 < p < ∞ on any Riemannian manifold with C r , r > 1, metric tensor.In conformal geometry, the case p = n is special since the class of n-harmonic functions is preserved under conformal transformations.This fact was used in [21] to study the regularity of such maps: if one fixes n-harmonic coordinate systems both in the domain and target manifold, the components of a conformal mapping will solve an elliptic equation.Thus nharmonic coordinates may be considered as a gauge condition for ellipticity in the regularity problem for conformal transformations.In any n-harmonic coordinate system, the following n-harmonic gauge condition is valid [21]: g kr g ka g kb g kk ∂ r g ab . (1.1) We now recall the definition of conformal curvature tensors.If (M, g) is an n-dimensional Riemannian manifold, write R abcd , R ab , and R for the Riemann curvature tensor, Ricci tensor, and scalar curvature, respectively.The Schouten tensor is defined as g ab .
The Weyl tensor of (M, g) is the 4-tensor W abcd = R abcd + P ac g bd − P bc g ad + P bd g ac − P ad g bc , the Cotton tensor is the 3-tensor and the Bach tensor is the 2-tensor We also consider the Fefferman-Graham obstruction tensor where ∆ = ∇ i ∇ i and n ≥ 4 is even.If n = 4 then O ab = B ab .These tensors have the following behavior under conformal scaling in various dimensions: [4], [7], [9], [12], [13], [27] for additional information on these tensors.
If T is one of the above tensors, the next result shows that the equation T (g) = h becomes elliptic under two natural local gauge conditions: one writes both sides of the equation in n-harmonic coordinates and considers the conformally normalized metric ĝjk = |g| −1/n g jk with determinant one.Here |g| = det (g jk ), and we use that n-harmonic coordinates for g are also n-harmonic coordinates for the conformal metric ĝ.
Theorem 1.1.Let (M, g) be a smooth n-dimensional Riemannian manifold.The Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors are elliptic operators in n-harmonic coordinates, in the sense that after applying both the n-harmonic gauge condition (1.1) and the condition |ĝ| = 1, the linearizations of the resulting operators at ĝ are elliptic.
After interpreting the conformal curvature tensors of low regularity metrics in a suitable sense, elliptic regularity results will follow readily.The regularity conditions will be given in terms of tensors 2n O ab which are invariant in conformal scaling g → cg.We use the notation C r * for the Zygmund spaces [25].
Theorem 1.2.Let M be a smooth n-dimensional manifold, and let g ∈ C r * in some system of local coordinates.
(a) If n ≥ 4, r > 1, and There are a number of results in conformal geometry that employ various gauges to obtain elliptic or parabolic regularity and existence results.The works [1], [2], [16], [27] involve constant scalar curvature gauges, sometimes obtained via the solution of the Yamabe problem [20], and study asymptotically locally Euclidean manifolds with obstruction flat metrics or boundary regularity of conformally compact Einstein metrics.In [6] a version of the DeTurck trick is used to show that certain analogues of the Ricci flow, including a modified flow involving the obstruction tensor, are locally well posed.In [14], [15] quadratic curvature functionals are studied with an emphasis on the structure of the space of their critical metrics.In these papers the corresponding Euler-Lagrange equations become elliptic via suitable gauge conditions.
Let us describe an argument from [2], [16], [27] for Bach flat metrics with n = 4. Choosing a conformal metric with constant scalar curvature, the equation of Bach flatness for the conformally scaled metric reads This becomes a fourth order elliptic equation in harmonic coordinates for the conformally scaled metric.This fact could be used to give another proof of an analogue of Theorem 1.2, if a solution to the (local) Yamabe problem for sufficiently low regularity metrics is provided.
We thus have two possible sets of gauge conditions for ellipticity of conformal curvature tensors: (n-harmonic coordinates + determinant one) and (constant scalar curvature + harmonic coordinates).Let us compare these briefly.The first approach works for C 1,α metrics with α > 0 and relies on an n-harmonic coordinate system.In [21] such coordinates (u 1 , . . ., u n ) near p were constructed by choosing some smooth coordinates (x 1 , . . ., x n ) near p and then solving for each j the Dirichlet problem An easy variational argument shows that this problem has a unique solution.The point is to show that (u 1 , . . ., u n ) is a C 1 diffeomorphism if ε is small enough, and this relies on standard regularity theory for quasilinear equations (essentially the fact that any p-harmonic function is C 1,α regular).Given n-harmonic coordinates, the conformal normalization (determinant one) is trivial.
The second approach based on a constant scalar curvature gauge requires solving the Yamabe problem (a semilinear equation with a constraint) at least in a small neighborhood; the global Yamabe problem is solvable for C 1,1 metrics (see [8,Section VII.7]), and we refer to [11], [17], [22] for versions of the local problem.The recent paper [17] deals with W 1,p , p > n, metrics.Given a conformal metric with constant scalar curvature, the other gauge condition involves a standard harmonic coordinate system.
A possible benefit of the n-harmonic coordinate approach is the similarity to harmonic coordinates in Riemannian geometry.In fact, it appears that n-harmonic coordinates are a natural generalization of harmonic coordinates in conformal geometry, and other Riemannian applications of harmonic coordinates might have conformal analogues.On the other hand, n-harmonic coordinates exist only locally, and if global gauge conditions are required then scalar curvature gauges are available via the Yamabe problem.Finally, n-harmonic coordinates exist at least for C 1,α metrics whereas global results for the Yamabe problem in the literature seem to require metrics with two derivatives.One motivation for this work was to find characterizations of local conformal flatness beyond the classical C 3 case, and it is an interesting question (also asked in [18, Section 2.7]) if one can have such characterizations for very low regularity metrics.
formulas.The Christoffel symbols are given by Noting the identity g ab ∂ l g ab = ∂ l (log|g|), we see that This also implies that The Riemann curvature tensor is given by Here The scalar curvature is R = g bc R bc .The Schouten, Weyl, Cotton and Bach tensors were defined in the introduction.By using Bianchi identities as in [27], the Cotton and Bach tensors can also be written as Below, we write T k for quantities that depend smoothly on components of the metric and their derivatives up to order k.We will also write for the principal parts of the Laplace-Beltrami operator and its square.
2.1.Bach tensor.All ellipticity results in this paper ultimately reduce to the fact that the Bach tensor is elliptic in n-harmonic coordinates when acting on metrics with determinant one.In the following, if T = T (g) is a nonlinear differential operator acting on metrics, we denote by σ(T ) the principal symbol of its linearization at g.
Lemma 2.1.The components of the Bach tensor satisfy in any local coordinate system in which |g| = 1.
Proof.Assume that the metric satisfies |g| = 1 in a given system of local coordinates.Then The Schouten tensor satisfies For the Bach tensor, we obtain Next we observe that This implies In the last step we used that Γ b = g km ∂ k g mb when |g| = 1, and Γ l = g lr Γ r .For the remaining terms in the Bach tensor, we use that Collecting these facts, we have proved the lemma.
Notice that in any harmonic coordinate system in which |g| = 1, the Bach tensor would have the form The operator on the right is obviously elliptic.However, in general it is not clear how to find harmonic coordinates with |g| = 1 even after conformal scaling, owing to the fact that the equations Γ k = 0 for harmonic coordinates are not conformally invariant.This is where n-harmonic coordinates become useful: they allow the conformal normalization |g| = 1, and the Bach tensor turns out to be elliptic in these coordinates.
Lemma 2.2.The components of the Bach tensor satisfy in any n-harmonic coordinate system with |g| = 1, where Γk is as in (1.1): g kr g ka g kb g kk ∂ r g ab .
Proof.Follows from Lemma 2.1 and the fact that Γ k = Γk in n-harmonic coordinates (see [21, Remark after Theorem 2.1]).
Consider the right hand side operator in Lemma 2.2 acting on symmetric positive definite matrix functions, and denote by Q = Q g the principal part of the linearization of this operator at g. Then Q is the principal part of the following linear operator, acting on symmetric matrix functions h by where L = L g = g ab ∂ ab , and Γk = Γk g is the operator g kr g ka g kb g kk ∂ r h ab .
Proof.The principal symbol of Q, acting on a symmetric matrix function h, is given by Here, the norms | • | and the raising and lowering of indices are taken with respect to the metric g.We also have Taking a = b, this gives If ξ = 0 and if q(ξ)h = 0, then also for all a = 1, . . ., n.This implies that h 11 g 11 = . . .= h nn g nn = λ for some function λ, and therefore Thus h aa = 0 for all a, so σ(−i Γa )(h) = 0 for all a and (q(ξ)h) ab = |ξ| 4 h ab = 0 for all a, b.It follows that h = 0, showing that Q is elliptic.
We are now ready to prove the main ellipticity result.
Proof of Theorem 1.1.Let (M, g) be a smooth Riemannian manifold, fix an n-harmonic coordinate system, and let ĝjk = |g| −1/n g jk in these coordinates.
Using the conformal invariance of the n-harmonic equation, our coordinate system is n-harmonic also with respect to the conformal metric ĝ.Let Bab be the Bach tensor for ĝ in n-harmonic coordinates after applying the nharmonic gauge condition (1.1) and the condition |ĝ| = 1.It follows from Lemmas 2.1-2.3 that the principal part of the linearization of Bab is elliptic in dimensions n ≥ 4.
Moving on to the Weyl tensor, the formula where Ŵ is the Weyl tensor of ĝ after applying the gauge conditions, and σ denotes the principal symbol of the linearization applied to a matrix function h.Thus σ( Ŵakbl )(ξ)h = 0 implies h = 0 by the ellipticity of Bab , showing that Ŵabcd is overdetermined elliptic (its principal symbol is injective).The ellipticity of Cotton and obstruction tensors follow in a similar way from the identities

Elliptic regularity results
We now consider low regularity Riemannian metrics, and establish elliptic regularity results for conformal curvature tensors.The Weyl tensor will be considered in detail and the arguments for the other tensors will be sketched.We first assume that in some system of local coordinates near a point.(In this section all function spaces are assumed to be of the local variety near a point, that is, we write W 1,2 instead of W 1,2 loc (U ) etc.) This seems to be a minimal assumption for defining the Weyl tensor: the set W 1,2 ∩ L ∞ is an algebra under pointwise multiplication [19], [5], and therefore g jk , |g| ∈ W 1,2 ∩ L ∞ .We see that g rs R rs g ab , we have Now, since multiplication by an W 1,2 ∩ L ∞ function maps W 1,q to W 1,2 for any q > n, it also maps W −1,2 to W −1,q ′ , and we have This shows that if g jk ∈ W 1,2 ∩L ∞ in some system of local coordinates, then one can make sense of the components W d abc of the Weyl tensor as elements in W −1,q ′ + L 1 in these coordinates for any q > n.It is easy to see that also R abcd , W abcd ∈ W −1,q ′ + L 1 in this case, and the identity W (cg) = cW (g) remains true if c, g jk ∈ W 1,2 ∩ L ∞ .
By similar arguments, if g jk ∈ W 2,2 ∩ W 1,∞ in some local coordinates, the components C abc of the Cotton tensor may be interpreted as elements of W −1,2 and one has 2 O(g) for n ≥ 4 even.We omit the details of these computations.
We will need the following elliptic regularity result, which is, after using suitable cutoffs and extensions, a consequence of [25,Theorem 14.4.2](the argument also applies to overdetermined elliptic operators).For completeness, the details are given in Appendix A. Proof.If g were sufficiently smooth, for instance C 4 , we could readily use the formula B ab = ∇ k ∇ l W akbl + 1 2 R kl W akbl to have ξ a ξ d σ(W abcd )h = −σ(B bc )h, which together with Lemma 2.1 would yield the claim.For C r * metrics, r > 1, we have to be more careful because the derivation of the different formulas for the Bach tensor uses the Bianchi identities that require more derivatives.We can however emulate the Bianchi identities on the symbol level using the equations